1. College Algebra Chapter 6 Matrices and Determinants and Applications
Section 6.4
Inverse Matrices and Matrix Equations

2. Concepts
1. Identify Identity and Inverse Matrices
2. Determine the Inverse of a Matrix
3. Solve Systems of Linear Equations Using the Inverse of a Matrix

3. Identify Identity and Inverse Matrices
The identity matrix In is the n  n square matrix with 1’s along the main diagonal and 0’s for all other elements.
Identity matrix of order 2 =
Identity matrix of order 3 =
For an n  n square matrix A:
(Identity property of matrix multiplication)

5. Identify Identity and Inverse Matrices
Let A be an n  n matrix and In be the identity matrix of order n. If there exists an n  n matrix A–1 such that
then A–1 is the multiplicative inverse of A.

6. Example 1:
Determine if
are inverses.

7. Example 1 continued:

9. Concepts
1. Identify Identity and Inverse Matrices
2. Determine the Inverse of a Matrix
3. Solve Systems of Linear Equations Using the Inverse of a Matrix

10. Determine the Inverse of a Matrix
Let A be an n  n matrix for which A–1 exists,
and let In be the n  n identity matrix.
To find A–1:
Step 1: Write a matrix of the form
Step 2: Perform row operations to write the matrix in the form
Step 3: The matrix B is A–1.

11. Determine the Inverse of a Matrix
Note: Not all matrices have a multiplicative inverse.
If a matrix A is reducible to a row-equivalent matrix with one or more rows of zeros, the matrix does not have an inverse, and we say that the matrix is singular.
A matrix that does have a multiplicative inverse is said to be invertible or nonsingular.

12. Example 2:
Given , find A-1 if possible.

13. Example 2 continued:

14. Example 3:
Given , find A-1 if possible.

18. Determine the Inverse of a Matrix
Formula for the inverse of a 2  2 invertible matrix:
Let be an invertible matrix.
Then the inverse is given by:

19. Example 4:
Given , find A-1.

21. Concepts
1. Identify Identity and Inverse Matrices
2. Determine the Inverse of a Matrix
3. Solve Systems of Linear Equations Using the Inverse of a Matrix

22. Solve Systems of Linear Equations Using the Inverse of a Matrix
A system of linear equations written in standard form can be represented by using matrix multiplication.
For example:
A ∙ X = B
corresponding matrix equation

23. Solve Systems of Linear Equations Using the Inverse of a Matrix
A ∙ X = B
coefficient
matrix
column matrix of variables
column matrix of constants

24. Solve Systems of Linear Equations Using the Inverse of a Matrix
AX = B To solve this equation,
the goal is to isolate X.
A–1AX = A–1B Multiply both sides by A–1
(provided that A–1 exists).
X = A–1B

25. Example 5:
Solve the system by using the inverse of the coefficient matrix.

26. Example 5 continued: