4.5 2x2 matrices, determinants and inverses 1.evaluating determinants of 2x2 matrices 2.using...
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4.52x2 Matrices, Determinants and
Inverses1. Evaluating Determinants of 2x2 Matrices
2. Using Inverse Matrices to Solve Equations
1) Evaluating Determinants of 2x2 Matrices
When you multiply two matrices together, in the order AB or BA, and the result is the identity matrix, then matrices A and B are inverses.
10
01I
Identity matrix
1) Evaluating Determinants of 2x2 Matrices
To show two matrices are inverses…
AB = I OR BA = I
AA-1 = I OR A-1A = I
Inverse of A Inverse of A
You only have to prove ONE of these.
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.
17
13A
3.07.0
1.01.0B
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.
17
13A
3.07.0
1.01.0B
3.07.0
1.01.0
17
13AB
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.
17
13A
3.07.0
1.01.0B
3.07.0
1.01.0
17
13AB
10
01AB
AB = I. Therefore, B is the inverse of A and A is the inverse of B.
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.
17
13A
3.07.0
1.01.0B
3.07.0
1.01.0
17
13AB
17
13
3.07.0
1.01.0BA
10
01AB
Check by multiplying BA…answer should be the same
AB = I. Therefore, B is the inverse of A and A is the inverse of B.
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Show that B is the multiplicative inverse of A.
17
13A
3.07.0
1.01.0B
3.07.0
1.01.0
17
13AB
17
13
3.07.0
1.01.0BA
10
01AB
10
01BA
Check by multiplying BA…answer should be the same
AB = I. Therefore, B is the inverse of A and A is the inverse of B.
1) Evaluating Determinants of 2x2 Matrices
Example 2:
Show that the matrices are multiplicative inverses.
83
52A
23
58B
1) Evaluating Determinants of 2x2 Matrices
Example 2:
Show that the matrices are multiplicative inverses.
83
52A
23
58B
83
52
23
58BA
10
01BA
BA = I. Therefore, B is the inverse of A and A is the inverse of B.
The determinant is used to tell us if an inverse exists.
If det ≠ 0, an inverse exists.
If det = 0, no inverse exists.
1) Evaluating Determinants of 2x2 Matrices
1) Evaluating Determinants of 2x2 Matrices
To calculate a determinant…
dc
baA dc
baA det
1) Evaluating Determinants of 2x2 Matrices
To calculate a determinant…
dc
baA dc
baA det
dc
ba Multiply along the diagonal
1) Evaluating Determinants of 2x2 Matrices
To calculate a determinant…
dc
baA dc
baA det
dc
ba
bcad
Multiply along the diagonal
Equation to find the determinant
1) Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.
95
87det
1) Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.
95
87det
95
87det
1) Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.
95
87det
95
87
95
87det
1) Evaluating Determinants of 2x2 Matrices
Example 1: Evaluate the determinant.
95
87det
95
87
)5)(8()9)(7(
23
det = -23
Therefore, there is an inverse.
95
87det
1) Evaluating Determinants of 2x2 Matrices
Example 2: Evaluate the determinant.
24
24det
1) Evaluating Determinants of 2x2 Matrices
Example 2: Evaluate the determinant.
24
24det
)2)(4()2)(4( 0
24
24det
1) Evaluating Determinants of 2x2 Matrices
Example 2: Evaluate the determinant.
24
24det
)2)(4()2)(4( 0
24
24det
det = 0
Therefore, there is no inverse.
1) Evaluating Determinants of 2x2 Matrices
How do you know if a matrix has an inverse AND what that inverse is?
ac
bd
AA
det
11
ac
bd
bcadA
11
Equations to find an inverse matrix
p.201
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
Step 1: Find det M
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
Step 1: Find det M
)5)(2()4)(2( bcad
2
det M = -2, the inverse of M exists.
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
Step 2: Rewrite the matrix in form.
ac
bd
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
Change signs
Step 2: Rewrite the matrix in form.
ac
bd
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
Change signs
?5
2?
Step 2: Rewrite the matrix in form.
ac
bd
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
Change positions
?5
2?
Step 2: Rewrite the matrix in form.
ac
bd
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
Step 2: Rewrite the matrix in form.
ac
bd
25
24
Change positions
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
Step 3: Use the equation to find the inverse.
25
24
25
24
2
11M
1) Evaluating Determinants of 2x2 Matrices
Example 1:
Determine whether the matrix has an inverse. If an inverse exists, find it.
45
22M
Step 3: Use the equation to find the inverse.
25
24
25
24
2
11M
15.2
121M
1) Evaluating Determinants of 2x2 Matrices
Example 2:
Determine whether the matrix has an inverse. If an inverse exists, find it.
31
42
1) Evaluating Determinants of 2x2 Matrices
Example 2:
Determine whether the matrix has an inverse. If an inverse exists, find it.
31
42
)1)(4()3)(2( bcad
2
31
42
31
42det
1) Evaluating Determinants of 2x2 Matrices
Example 2:
Determine whether the matrix has an inverse. If an inverse exists, find it.
31
42
21
43
2
11A
15.0
25.11A
Homework
p.203 #1, 2, 4, 5, 14, 15, 27, 28, 32, 34
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