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

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

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)

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.

Example 1: Determine if are inverses.

Example 1 continued:

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

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.

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.

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

Example 2 continued:

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

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:

Example 4: Given , find A-1.

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

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

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

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

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

Example 5 continued: