2. Chapter 7: Matrices and Systems of Equations and Inequalities
7.2 Solution of Linear Systems in Three Variables
7.3 Solution of Linear Systems by Row Transformations
7.4 Matrix Properties and Operations
7.5 Determinants and Cramer’s Rule
7.6 Solution of Linear Systems by Matrix Inverses
7.7 Systems of Inequalities and Linear Programming
7.8 Partial Fractions

3. 7.6 Solution of Linear Systems by Matrix Inverses
If there is a multiplicative identity matrix I, such that
for any matrix A, then A and I must be square matrices of the same dimensions.
The 2 × 2 identity matrix, denoted I2, is
This is easily verified by showing A I2 = A and
I2 A = A, for any 2 × 2 matrix A.

4. 7.6 Using the 3 × 3 Identity Matrix I3
Example Show that
Graphing Calculator Solution Using

5. 7.6 Multiplicative Inverses of Square Matrices
Suppose If AB = I2 and BA = I2, then B
is the inverse of A.
Inverse of a 2 × 2 matrix

6. 7.6 The Inverse of a 2 × 2 Matrix
Solving the system
yields
We can show that AB = I2 and BA = I2.. Thus, we can conclude
that B is the inverse of A, written A-1, provided that the det A  0.

7. 7.6 The Inverse of a 2 × 2 Matrix
If and det A  0, then or

8. 7.6 Finding the Inverse of a 2 × 2 Matrix
Example Find A-1 if it exists.
Analytic Solution
(a)
(b)
Here, A-1 does not exist.

9. 7.6 Finding the Inverse of a 2 × 2 Matrix
Graphing Calculator Solution
(a)
(b)
The calculator returns a singular matrix error when directed to find the inverse of a matrix whose determinant is 0.

10. 7.6 Solving Linear Systems Using Inverse Matrices
Solve the system AX = B, where A is the coefficient matrix, X is the matrix of variables, and B is the matrix of the constants.
Note: When multiplying by matrices on both sides, multiply in the same order on both sides.
Multiply both sides by A-1.
Associative property
Multiplicative inverse property
Identity property

11. 7.6 Solving Linear Systems Using Inverse Matrices
Example Solve the system using the inverse of the
coefficient matrix.
Analytic Solution

12. 7.6 Solving Linear Systems Using Inverse Matrices
The solution set {(x, y, z)} = {(4, 2, –3)}.

13. 7.6 Solving Linear Systems Using Inverse Matrices
Graphing Calculator Solution
Enter the coefficient matrix A and the constant matrix B.
Make sure the det A  0.
The solution verifies the results achieved analytically.