mathh 1300: section 4- 4 matrices: basic operations
DESCRIPTION
TRANSCRIPT
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Math 1300 Finite MathematicsSection 4.4 Matrices: Basic Operations
Jason Aubrey
Department of MathematicsUniversity of Missouri
June 20, 2011
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Two matrices are equal if they have the same size andtheir corresponding elements are equal.
For example,1 3 24 −1 −21 0 −1
=
1 3 24 −1 −21 0 −1
The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.Addition is not defined for matricies of different sizes.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Two matrices are equal if they have the same size andtheir corresponding elements are equal. For example,1 3 2
4 −1 −21 0 −1
=
1 3 24 −1 −21 0 −1
The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.Addition is not defined for matricies of different sizes.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Two matrices are equal if they have the same size andtheir corresponding elements are equal. For example,1 3 2
4 −1 −21 0 −1
=
1 3 24 −1 −21 0 −1
The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.
Addition is not defined for matricies of different sizes.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Two matrices are equal if they have the same size andtheir corresponding elements are equal. For example,1 3 2
4 −1 −21 0 −1
=
1 3 24 −1 −21 0 −1
The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.Addition is not defined for matricies of different sizes.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Evaluate1 1 24 −3 −21 0 −4
+
−2 3 24 −1 −20 0 −4
=
−1 4 48 −4 −41 0 −8
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Evaluate1 1 24 −3 −21 0 −4
+
−2 3 24 −1 −20 0 −4
=
−1 4 48 −4 −41 0 −8
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Evaluate
[5 0 −21 −3 8
]+
−1 70 6−2 8
Undefined!
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Evaluate
[5 0 −21 −3 8
]+
−1 70 6−2 8
Undefined!
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, then
A + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted −M is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 −3 22 −4 3
]then −M =
[−1 3 −2−2 4 −3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)
(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted −M is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 −3 22 −4 3
]then −M =
[−1 3 −2−2 4 −3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted −M is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 −3 22 −4 3
]then −M =
[−1 3 −2−2 4 −3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted −M is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 −3 22 −4 3
]then −M =
[−1 3 −2−2 4 −3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted −M is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 −3 22 −4 3
]then −M =
[−1 3 −2−2 4 −3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted −M is a matrix withelements that are the negatives of the elements in M.
For
example, if M =
[1 −3 22 −4 3
]then −M =
[−1 3 −2−2 4 −3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted −M is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 −3 22 −4 3
]then −M =
[−1 3 −2−2 4 −3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A and B are matrices of the same size, we define subtractionas follows:
A − B = A + (−B)
For example,
[3 1 2−3 3 1
]−[
2 2 −3−1 1 0
]=
[1 −1 5−2 2 1
]
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A and B are matrices of the same size, we define subtractionas follows:
A − B = A + (−B)
For example,
[3 1 2−3 3 1
]−[
2 2 −3−1 1 0
]=
[1 −1 5−2 2 1
]
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product of a number k and a matrix M, denoted kM, is amatrix formed by multiplying each element of M by k .
For example,
3
−1 04 3−2 1/3
=
−3 012 9−6 1
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product of a number k and a matrix M, denoted kM, is amatrix formed by multiplying each element of M by k .
For example,
3
−1 04 3−2 1/3
=
−3 012 9−6 1
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Examples: Perform the indicated operations, if possible, giventhat
A =
1 41 33 −7
and B =
0 1−2 24 1
.
3A + B
3 123 99 −21
+
0 1−2 24 1
=
3 131 11
13 −20
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Examples: Perform the indicated operations, if possible, giventhat
A =
1 41 33 −7
and B =
0 1−2 24 1
.
3A + B 3 123 99 −21
+
0 1−2 24 1
=
3 131 11
13 −20
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Examples: Perform the indicated operations, if possible, giventhat
A =
1 41 33 −7
and B =
0 1−2 24 1
.
3A + B 3 123 99 −21
+
0 1−2 24 1
=
3 131 11
13 −20
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A − B
1 41 33 −7
+
0 −12 −2−4 −1
=
1 33 1−1 −8
−A + 2B
−1 −4−1 −3−3 7
+
0 2−4 48 2
=
−1 −2−5 15 9
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A − B 1 41 33 −7
+
0 −12 −2−4 −1
=
1 33 1−1 −8
−A + 2B
−1 −4−1 −3−3 7
+
0 2−4 48 2
=
−1 −2−5 15 9
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A − B 1 41 33 −7
+
0 −12 −2−4 −1
=
1 33 1−1 −8
−A + 2B
−1 −4−1 −3−3 7
+
0 2−4 48 2
=
−1 −2−5 15 9
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A − B 1 41 33 −7
+
0 −12 −2−4 −1
=
1 33 1−1 −8
−A + 2B −1 −4
−1 −3−3 7
+
0 2−4 48 2
=
−1 −2−5 15 9
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A − B 1 41 33 −7
+
0 −12 −2−4 −1
=
1 33 1−1 −8
−A + 2B −1 −4
−1 −3−3 7
+
0 2−4 48 2
=
−1 −2−5 15 9
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
DefinitionThe product of a 1× n row matrix and an n × 1 column matrix isa 1 × 1 matrix given by:
[a1 a2 . . . an]
b1b2...
bn
= [a1b1 + a2b2 + · · ·+ anbn]
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
DefinitionThe product of a 1× n row matrix and an n × 1 column matrix isa 1 × 1 matrix given by:
[a1 a2 . . . an]
b1b2...
bn
= [a1b1 + a2b2 + · · ·+ anbn]
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example:
[1 2 −2
] 021
= [1(0) + 2(2)− 2(1)] = [2]
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example:
[1 2 −2
] 021
= [1(0) + 2(2)− 2(1)]
= [2]
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example:
[1 2 −2
] 021
= [1(0) + 2(2)− 2(1)] = [2]
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Definition (General Matrix Product)If A is an m × p matrix and B is a p × n matrix, the matrixproduct of A and B, denoted AB, is an m × n matrix whoseelements in the i th row and j th column is the real numberobtained from the product of the i th row of A and the j th columnof B. If the number of columns in A does not equal the numberof rows in B, the matrix product AB is not defined.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size
3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3.
However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size
1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.
The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size
2 × 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3.
However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA
is undefined.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals
3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0
− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals
3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0
− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8)
3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0
− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8)
3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0
− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13
1 73 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2)
3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0
− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13
1 73 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2)
3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0
− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1
73 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1
73 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 7
3 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0
3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 7
3 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0
3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73
− 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0
3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73
− 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0
3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3
96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0
− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3
96 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0
− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 9
6 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8)
− 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 9
6 0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8)
− 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 96
0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2)
− 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 96
0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2)
− 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 96 0
0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 96 0
0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 −23 1 0−2 1 1
1 −1 20 0 38 −2 1
This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)
3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)
=
− 13 1 73 − 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics