1. 3.8 Use Inverse Matrices to Solve Linear Systems

2. The number 1
Is the multiplicative identity for real numbers because 1*a=a and a*1=a for all real numbers a.
For matrices, the n x n identity matrix is the matrix that has 1’s on the main diagonal and 0’s elsewhere.
2 x 2 Identity Matrix
3 x 3 Identity Matrix
If A is any n x n matrix and I is the n x n identity matrix, then IA=A and AI=A

3. Two n x n matrices are inverses of each other if their product (in both orders) is the n x n identity matrix.
For example, matrices A and B about to appear are inverses of each other.
We will use this symbol to denote the inverse of matrix A

4. The Inverse of a 2 x 2 Matrix
The inverse of matrix
Provided ad – cb is not equal to zero.

5. A I = I A = A
Verify the identity property above indicated for matrix A below: 8 2 6 4 A= 1 I=
Diagonal 1 8 2 6 4
I A = =
1(8) + 0(6)
1(2) + 0(4)
0(8) + 1(6)
0(2) + 1(4) 8 2 6 4 = 1 8 2 6 4
A I = =
8(1) + 6(0)
2(1) + 4(0)
8(0) + 6(1)
2(0) + 4(1) 8 2 6 4 =

6. Calculate the A for matrix A:-1 8 2 6 4 A= 8 2 6 4
= (8)(4) –(6)(2)
=32 -12
=20 = 4 20 -2 -6 8 = 1 5 -1 10 -3 2 4 -2 -6 8 1 20
A = -1

7. Calculate the A for matrix A:-1 4 3 5 1 A= 4 3 5 1
= (4)(1) –(5)(3)
=4 -15
= -11 = -1 11 3 5 -4 1 -3 -5 4 1
-11
A = -1

8. Write the system of equations represented by each matrix equation:-3 6 7 1 x y = 15 -8
Notice that this matrix equation could be solved for X by multiplying both sided of the equation by the Inverse Matrix A
-3x + 6y = 15
7x + y = -8 5 9 -2 4 x y =
5x + 9y = 0
-2x + 4y = 5

9. Multiplying both sides by the inverse:
Solve the following system of equations using a Matrix Equation:
4x + 2y = 10
5x + y = 17 1
Multiplying both sides by the inverse:
Write as matrix equation: -1 6 1 3 5 -2 = -1 6 1 3 5 -2 4 2 5 1 x y = 10 17 4 2 5 1 x y 10 17
Finding the determinant of the
coefficient matrix: 4 2 5 1
= (4)(1) –(5)(2)
=4 -10
= -6
Finding the inverse of the coefficient matrix: = -1 6 2 5 -4 = -1 6 1 3 5 -2 1 -2 -5 4 1 -6

10. 4x + 2y = 10
5x + y = 17 1
Write as matrix equation: -1 6 1 3 5 -2 -1 6 1 3 5 -2 4 2 5 1 x y 10 17 4 2 5 1 x y 10 17 = = -1 6 1 3
(4)
(5) + -1 6 1 3
(2)
(1) + -1 6 1 3
(10)
(17) + x y = 5 6 -2 3
(10)
(17) + 5 6 -2 3
(4)
(5) + 5 6 -2 3
(2)
(1) + 1 x y = 4 -3 x y = 4 -3
Solution is (4,-3)

11. Multiplying both sides by the inverse:
Solve the following system of equations using matrices:
2x + 5y = 13
6x + 3y = 3
Multiplying both sides by the inverse:
Write as matrix equation: -1 8 5 24 1 4 12 = -1 8 5 24 1 4 12 2 5 6 3 x y = 13 3 2 5 6 3 x y 13 3
Finding the determinant of the
coefficient matrix: 2 5 6 3
= (2)(3) –(6)(5)
=6 -30
= -24
Finding the inverse of the coefficient matrix: = -3 24 5 6 -2 = -1 8 5 24 1 4 12 3 -5 -6 2 1
-24

12. Solve the following system of equations using matrices:
2x + 5y = 13
6x + 3y = 3
Write as matrix equation: -1 8 5 24 1 4 12 -1 8 5 24 1 4 12 2 5 6 3 x y 13 3 2 5 6 3 x y 13 3 = = -1 8 5 24
(2)
(6) + -1 8 5 24
(5)
(3) + -1 8 5 24
(13)
( 3) + = x y 1 4 -1 12
(13)
( 3) + 1 4 -1 12
(2)
(6) + 1 4 -1 12
(5)
(3) + 1 x y = -1 3 x y = -1 3
Solution is (-1,3)