4.5, 4.6 2 x 2 and 3 x 3 Matrices, Determinants, and Inverses Date: _____________.
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Transcript of 4.5, 4.6 2 x 2 and 3 x 3 Matrices, Determinants, and Inverses Date: _____________.
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4.5, 4.62 x 2 and 3 x 3 Matrices, Determinants, and
Inverses
Date: _____________
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Matrices are multiplicative inverses
• Page 199 – 2 definitions• Multiplicative Identity Matrix– Must be a square matrix, 2 x 2, 3 x 3, 4 x 4, etc. – Has 1’s in the main diagonal and 0’s elsewhere
• Multiplicative Inverse of a Matrix– when multiplying a matrix by its inverse, we get the
identity matrix1 A A I 1 A A I
1 0 01 0
, 0 1 00 1
0 0 1
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Use your calculator
Matrices are multiplicative inverses
2 5 8 5 ,
3 8 3 2
2 5
3 8A
8 5
3 2B
Show that these two matrices are multiplicative inverses
2 5 8 5
3 8 3 2A B
1 0
0 1
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Objective - To evaluate the determinates of 2 x 2 and 3 x 3 matrices.
2 2 Matrix
2 6 det
9 7
2 6
9 72(7)
2 6
9 7
Determinant can be labeled either way
Find the Determinant
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2 2 Matrix
2 6 det
9 7
2 6
9 72(7)
2 6
9 7
Determinant
9(6)
14 54
40a b
c dad cb
Objective - To evaluate the determinates of 2 x 2 and 3 x 3 matrices.
Find the Determinant
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Evaluate the Determinant for each Matrix
1 31.
5 2
3 72.
2 9
1(2) 3( 5) 13
2 23.
2 2
3(9) ( 2)7 41
( 2)(2) ( 2)(2) 0When the determinant = 0, then that matrix has NO INVERSE
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4 2 5
8 1 6
2 5 1
4 2 5
8 1 6
2 5 1
4
8
2
2
1
5
DeterminantTake the first 2 columns and rewrite them outside
Find the determinant of each 3x3 Matrix.
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2 5 4 2
8 6 8
4
12 5 2
1 1
5
4
Find the determinant of each 3x3 Matrix.
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4 5 4 2
8 1 8 1
2 25
2
6
1 5
244
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4 2 4 2
8 1 6 1
2 5 1 2
5
8
5
( )200 244
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4 2 4 2
8 6 8
5
1
2
1
5 1 2 5
0(1( )200 244
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4 2 5 2
8 1 8 1
2 2
4
6
5 1 5
120 0(1( )200 244
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4 2 5 4
8 1 6 1
2 5 1 2
2
8
5
)16
228 146 82
Solve
1 3 2
2 4 6
3 1 9
120 0(1( )200 244
Fun?Use your Calculator
1 3 2
2 4 6
3 1 9
A
Matrix, over to MATH, then det(, then go to Matrix, we want matrix A
det( ) 86A
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Determinant and its useThe determinant is used to find our inverseWe will use our calculator to find the inverse. Type in:
2 3
1 1A
2 3
det1 1
Find the determinant first:
(2)(1) (3)(1) 1
Therefore, it has an inverse
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Determinant and its useThe determinant is used to find our inverseWe will use our calculator to find the inverse. Type in:
2 3
1 1A
1A
1 3
1 2
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3 2
11
2
A 1 4
2 6
Find the inverse of the matrix
1A
1 2 1
1.5 3 1.75
0 1 0.5
A
1 0 2
3 2 1
6 4 0
1A
If A didn’t have an inverse, you’d get the message ERR: SINGULAR MAT
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Checking your answers.
AA 1 1 0
0 1
If you multiply inverses, you will always get the identity matrix.
This is a way you can check your answers
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Solve for X.
Linear EquationsMatrix Equations
AX Bax b a a
bx
a
1 1A AX A B 1IX A B
1X A B
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Objective - To solve systems using inverse matrices.
4 6 1
3 7 2X
4 6
3 7A
1
2B
1X A B1.9
1.1
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Do this one on your own to see if you understand
6 10 13 84
4 2 7 18
0 9 8 56
X
5
8
2
6 10 13
4 2 7
0 9 8
A
84
18
56
B
1X A B