section 2.2 and 2.3 the inverse of a matrix
TRANSCRIPT
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Section 2.2 and 2.3 The Inverse of a Matrix
Gexin [email protected]
College of William and Mary
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 2: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/2.jpg)
Elementary Matrices
If A is invertible, what is the reduced echelon form of A?
If A is invertible, then A can be row reduced to an identity matrix.
We now find A−1 by watching the row reduction of A to I .
An elementary matrix is one that is obtained by performing a singleelementary row operation on an identity matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 3: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/3.jpg)
Elementary Matrices
If A is invertible, what is the reduced echelon form of A?
If A is invertible, then A can be row reduced to an identity matrix.
We now find A−1 by watching the row reduction of A to I .
An elementary matrix is one that is obtained by performing a singleelementary row operation on an identity matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 4: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/4.jpg)
Elementary Matrices
If A is invertible, what is the reduced echelon form of A?
If A is invertible, then A can be row reduced to an identity matrix.
We now find A−1 by watching the row reduction of A to I .
An elementary matrix is one that is obtained by performing a singleelementary row operation on an identity matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 5: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/5.jpg)
Elementary Matrices
If A is invertible, what is the reduced echelon form of A?
If A is invertible, then A can be row reduced to an identity matrix.
We now find A−1 by watching the row reduction of A to I .
An elementary matrix is one that is obtained by performing a singleelementary row operation on an identity matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 6: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/6.jpg)
Ex. Compute E1A,E2A,E3A, and describe how these product can beobtained by elementary row operations on A, where
A =
a b vd e fg h i
and
E1 =
1 0 00 1 0−4 0 1
E2 =
0 1 01 0 00 0 1
E3 =
1 0 00 1 00 0 5
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
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Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 8: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/8.jpg)
If an elementary row operation is performed on an m × n matrix A,the resulting matrix can be written as EA, where the m×m matrix Eis the corresponding elementary matrix.
Each elementary matrix E is invertible. The inverse of E is theelementary matrix of the same type that transforms E back into I .
if E =
1 0 0 0d 1 0 00 0 1 00 0 0 1
, then E−1 =
1 0 0 0−d 1 0 00 0 1 00 0 0 1
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
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If an elementary row operation is performed on an m × n matrix A,the resulting matrix can be written as EA, where the m×m matrix Eis the corresponding elementary matrix.
Each elementary matrix E is invertible. The inverse of E is theelementary matrix of the same type that transforms E back into I .
if E =
1 0 0 0d 1 0 00 0 1 00 0 0 1
, then E−1 =
1 0 0 0−d 1 0 00 0 1 00 0 0 1
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 10: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/10.jpg)
If an elementary row operation is performed on an m × n matrix A,the resulting matrix can be written as EA, where the m×m matrix Eis the corresponding elementary matrix.
Each elementary matrix E is invertible. The inverse of E is theelementary matrix of the same type that transforms E back into I .
if E =
1 0 0 0d 1 0 00 0 1 00 0 0 1
, then E−1 =
1 0 0 0−d 1 0 00 0 1 00 0 0 1
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 11: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/11.jpg)
Theorem 7: An n × n matrix A is invertible if and only if A is rowequivalent to In; furthermore, any sequence of elementary row operationsthat reduces A to In also transforms In into A−1.
PF. Suppose A is equivalent to In.
Then there are elementary matrices E1,E2, . . . ,Ep so thatEp . . .E2E1A = In.
As Ei s are invertible, their product is also invertible, so we have
(Ep . . .E2E1)−1(Ep . . .E2E1)A = (Ep . . .E2E1)−1In
So we have A = (Ep . . .E1)−1.
It follows that A−1 = ((Ep . . .E1)−1)−1 = Ep . . .E1.
Then A−1 = Ep . . .E1, which says that A−1 results from applyingE1, . . . ,Ep successively to In.
This is the same sequence that reduced A to In.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 12: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/12.jpg)
Theorem 7: An n × n matrix A is invertible if and only if A is rowequivalent to In; furthermore, any sequence of elementary row operationsthat reduces A to In also transforms In into A−1.
PF. Suppose A is equivalent to In.
Then there are elementary matrices E1,E2, . . . ,Ep so thatEp . . .E2E1A = In.
As Ei s are invertible, their product is also invertible, so we have
(Ep . . .E2E1)−1(Ep . . .E2E1)A = (Ep . . .E2E1)−1In
So we have A = (Ep . . .E1)−1.
It follows that A−1 = ((Ep . . .E1)−1)−1 = Ep . . .E1.
Then A−1 = Ep . . .E1, which says that A−1 results from applyingE1, . . . ,Ep successively to In.
This is the same sequence that reduced A to In.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 13: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/13.jpg)
Algorithm to find A−1
If we place A and I side-by-side to form an augmented matrix[A I
],
then row operations on this matrix produce identical operations on Aand on I .
By Theorem 7, either there are row operations that transform A to Inand In to A−1, or else A is not invertible.
That is, [A I
]→[I A−1
]
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 14: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/14.jpg)
Algorithm to find A−1
If we place A and I side-by-side to form an augmented matrix[A I
],
then row operations on this matrix produce identical operations on Aand on I .
By Theorem 7, either there are row operations that transform A to Inand In to A−1, or else A is not invertible.
That is, [A I
]→[I A−1
]
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 15: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/15.jpg)
Algorithm to find A−1
If we place A and I side-by-side to form an augmented matrix[A I
],
then row operations on this matrix produce identical operations on Aand on I .
By Theorem 7, either there are row operations that transform A to Inand In to A−1, or else A is not invertible.
That is, [A I
]→[I A−1
]
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 16: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/16.jpg)
Algorithm to find A−1—-Example
Ex. Find the inverse of the matrix A =
0 1 01 0 34 −3 8
, if it exists.
Soln.[A I
]=
0 1 0 1 0 01 0 3 0 1 04 −3 8 0 0 1
→1 0 0 −9/2 7 −3/2
0 1 0 −2 4 −10 0 1 3/2 −2 1/2
.
So A−1 =
−9/2 7 −3/2−2 4 −13/2 −2 1/2
.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 17: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/17.jpg)
Algorithm to find A−1—-Example
Ex. Find the inverse of the matrix A =
0 1 01 0 34 −3 8
, if it exists.
Soln.[A I
]=
0 1 0 1 0 01 0 3 0 1 04 −3 8 0 0 1
→1 0 0 −9/2 7 −3/2
0 1 0 −2 4 −10 0 1 3/2 −2 1/2
.
So A−1 =
−9/2 7 −3/2−2 4 −13/2 −2 1/2
.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 18: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/18.jpg)
Algorithm to find A−1—-Example
Ex. Find the inverse of the matrix A =
0 1 01 0 34 −3 8
, if it exists.
Soln.[A I
]=
0 1 0 1 0 01 0 3 0 1 04 −3 8 0 0 1
→1 0 0 −9/2 7 −3/2
0 1 0 −2 4 −10 0 1 3/2 −2 1/2
.
So A−1 =
−9/2 7 −3/2−2 4 −13/2 −2 1/2
.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 19: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/19.jpg)
Invertible Linear Transformations
Matrix multiplication corresponds to composition of lineartransformations.
When a matrix A is invertible, the equation A−1Ax = x can beviewed as a statement about linear transformations.
See the following figure.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
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Invertible Linear Transformations
Matrix multiplication corresponds to composition of lineartransformations.
When a matrix A is invertible, the equation A−1Ax = x can beviewed as a statement about linear transformations.
See the following figure.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 21: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/21.jpg)
Invertible Linear Transformations
Matrix multiplication corresponds to composition of lineartransformations.
When a matrix A is invertible, the equation A−1Ax = x can beviewed as a statement about linear transformations.
See the following figure.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 22: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/22.jpg)
A linear transformation T : Rn → Rn is said to be invertible if thereexists a function S : Rn → Rn such that
S(T (x)) = x for all x in Rn (1)
T (S(x)) = x for all x in Rn (2)
Theorem 9: Let T : Rn → Rn be a linear transformation and let A bethe standard matrix for T . Then T is invertible if and only if A is aninvertible matrix. In that case, the linear transformation S given byS(x) = A−1x is the unique function satisfying equation (1) and (2).
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
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A linear transformation T : Rn → Rn is said to be invertible if thereexists a function S : Rn → Rn such that
S(T (x)) = x for all x in Rn (1)
T (S(x)) = x for all x in Rn (2)
Theorem 9: Let T : Rn → Rn be a linear transformation and let A bethe standard matrix for T . Then T is invertible if and only if A is aninvertible matrix. In that case, the linear transformation S given byS(x) = A−1x is the unique function satisfying equation (1) and (2).
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 24: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/24.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.(a) A is an invertible matrix.
(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 25: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/25.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.
(c) A has n pivot positions.(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 26: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/26.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.
(a) A is an invertible matrix.
(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.
(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 27: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/27.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.
(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.
(c) A has n pivot positions.(d) The equation Ax = 0 has only the trivial solution.
(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 28: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/28.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.
(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.
(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.
(f) The linear transformation x → Ax is one-to-one.(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 29: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/29.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.
(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.
(c) A has n pivot positions.
(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.
(f) The linear transformation x → Ax is one-to-one.
(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 30: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/30.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.
(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.
(c) A has n pivot positions.
(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.
(g) The equation Ax = b has at least one solution for each b in Rn.
(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 31: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/31.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.
(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.
(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.
(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 32: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/32.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.
(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.
(g) The equation Ax = b has at least one solution for each b in Rn.
(h) The columns of A span Rn.
(i) The linear transformation x → Ax maps Rn onto Rn.
(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 33: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/33.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.(a) A is an invertible matrix.
(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.
(d) The equation Ax = 0 has only the trivial solution.
(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.
(j) There is an n × n matrix C such that CA = I .
(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 34: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/34.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.(a) A is an invertible matrix.
(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.
(g) The equation Ax = b has at least one solution for each b in Rn.
(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .
(k) There is an n × n matrix D such that AD = I .
(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 35: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/35.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.
(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.
(c) A has n pivot positions.
(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .
(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 36: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/36.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.(a) A is an invertible matrix.
(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 37: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/37.jpg)
The Invertible Matrix Theorem
Theorem 8: Let A be a square n × n matrix. Then the followingstatements are equivalent.(a) A is an invertible matrix.(b) A is row equivalent to the n × n identity matrix.(c) A has n pivot positions.(d) The equation Ax = 0 has only the trivial solution.(e) The columns of A form a linearly independent set.(f) The linear transformation x → Ax is one-to-one.(g) The equation Ax = b has at least one solution for each b in Rn.(h) The columns of A span Rn.(i) The linear transformation x → Ax maps Rn onto Rn.(j) There is an n × n matrix C such that CA = I .(k) There is an n × n matrix D such that AD = I .(l) The equation Ax = b has a unique solution for each b in Rn
(m) AT is an invertible matrix.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 38: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/38.jpg)
The following fact follows from Theorem 8:
Let A and B be square matrices. If AB = I , then A and Bare both invertible, with B = A−1 and A = B−1.
The Invertible Matrix Theorem divides the set of all n × n matricesinto two disjoint classes: the invertible (nonsingular) matrices, andthe noninvertible (singular) matrices.
Each statement in the theorem describes a property of every n × ninvertible matrix.
The negation of a statement in the theorem describes a property ofevery n × n singular matrix.
For instance, an n × n singular matrix is not row equivalent to In,does not have n pivot position, and has linearly dependent columns.
The Invertible Matrix Theorem applies only to square matrices.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 39: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/39.jpg)
The following fact follows from Theorem 8:
Let A and B be square matrices. If AB = I , then A and Bare both invertible, with B = A−1 and A = B−1.
The Invertible Matrix Theorem divides the set of all n × n matricesinto two disjoint classes: the invertible (nonsingular) matrices, andthe noninvertible (singular) matrices.
Each statement in the theorem describes a property of every n × ninvertible matrix.
The negation of a statement in the theorem describes a property ofevery n × n singular matrix.
For instance, an n × n singular matrix is not row equivalent to In,does not have n pivot position, and has linearly dependent columns.
The Invertible Matrix Theorem applies only to square matrices.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 40: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/40.jpg)
The following fact follows from Theorem 8:
Let A and B be square matrices. If AB = I , then A and Bare both invertible, with B = A−1 and A = B−1.
The Invertible Matrix Theorem divides the set of all n × n matricesinto two disjoint classes: the invertible (nonsingular) matrices, andthe noninvertible (singular) matrices.
Each statement in the theorem describes a property of every n × ninvertible matrix.
The negation of a statement in the theorem describes a property ofevery n × n singular matrix.
For instance, an n × n singular matrix is not row equivalent to In,does not have n pivot position, and has linearly dependent columns.
The Invertible Matrix Theorem applies only to square matrices.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 41: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/41.jpg)
The following fact follows from Theorem 8:
Let A and B be square matrices. If AB = I , then A and Bare both invertible, with B = A−1 and A = B−1.
The Invertible Matrix Theorem divides the set of all n × n matricesinto two disjoint classes: the invertible (nonsingular) matrices, andthe noninvertible (singular) matrices.
Each statement in the theorem describes a property of every n × ninvertible matrix.
The negation of a statement in the theorem describes a property ofevery n × n singular matrix.
For instance, an n × n singular matrix is not row equivalent to In,does not have n pivot position, and has linearly dependent columns.
The Invertible Matrix Theorem applies only to square matrices.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 42: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/42.jpg)
The following fact follows from Theorem 8:
Let A and B be square matrices. If AB = I , then A and Bare both invertible, with B = A−1 and A = B−1.
The Invertible Matrix Theorem divides the set of all n × n matricesinto two disjoint classes: the invertible (nonsingular) matrices, andthe noninvertible (singular) matrices.
Each statement in the theorem describes a property of every n × ninvertible matrix.
The negation of a statement in the theorem describes a property ofevery n × n singular matrix.
For instance, an n × n singular matrix is not row equivalent to In,does not have n pivot position, and has linearly dependent columns.
The Invertible Matrix Theorem applies only to square matrices.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 43: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/43.jpg)
The following fact follows from Theorem 8:
Let A and B be square matrices. If AB = I , then A and Bare both invertible, with B = A−1 and A = B−1.
The Invertible Matrix Theorem divides the set of all n × n matricesinto two disjoint classes: the invertible (nonsingular) matrices, andthe noninvertible (singular) matrices.
Each statement in the theorem describes a property of every n × ninvertible matrix.
The negation of a statement in the theorem describes a property ofevery n × n singular matrix.
For instance, an n × n singular matrix is not row equivalent to In,does not have n pivot position, and has linearly dependent columns.
The Invertible Matrix Theorem applies only to square matrices.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix
![Page 44: Section 2.2 and 2.3 The Inverse of a Matrix](https://reader030.vdocument.in/reader030/viewer/2022011822/61d5ebc896e08628ec297de3/html5/thumbnails/44.jpg)
The following fact follows from Theorem 8:
Let A and B be square matrices. If AB = I , then A and Bare both invertible, with B = A−1 and A = B−1.
The Invertible Matrix Theorem divides the set of all n × n matricesinto two disjoint classes: the invertible (nonsingular) matrices, andthe noninvertible (singular) matrices.
Each statement in the theorem describes a property of every n × ninvertible matrix.
The negation of a statement in the theorem describes a property ofevery n × n singular matrix.
For instance, an n × n singular matrix is not row equivalent to In,does not have n pivot position, and has linearly dependent columns.
The Invertible Matrix Theorem applies only to square matrices.
Gexin Yu [email protected] Section 2.2 and 2.3 The Inverse of a Matrix