2x2 matrices, determinants and inverses 1.evaluating determinants of 2x2 matrices 2.using inverse...
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2x2 Matrices, Determinants and Inverses
1. 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 for multiplication
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. A Matrix with a determinant of zero is called a SINGULAR matrix
1) Evaluating Determinants of 2x2 Matrices
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
Take the product of the leading diagonal, and subtract the product of the non-leading diagonal
Equation to find the determinant
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
)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?
Given , the inverse of A is given by:
ac
bd
AA
det
11
Equation to find an inverse matrix
This is called the adjoint matrix. It is formed by interchanging elements in the leading diagonal and negating elements in the non-leading diagonal
dc
baA
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: Find the adjoint matrix. i.e
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: Find the adjoint matrix. i.e
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: Find the adjoint matrix. i.e
ac
bd
Adjoint of M
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: Find the adjoint matrix. i.e
ac
bd
Adjoint of 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 2: Find the adjoint matrix. i.e
ac
bd
25
24
Change positions
Adjoint of 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 3: Use the equation to find the inverse.
25
24
2
11M
ofMAdjoM
M intdet
11
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
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