Chapter 2 Systems of Linear Equations and Matrices Section 2.5 Matrix Inverses

What is a Matrix Inverse? The inverse of a matrix is comparable to the reciprocal of a real number. The product of a matrix and its identity matrix is always the matrix itself. In other words, multiplying a matrix by its identity matrix is like multiplying a number by 1.

Multiplicative Identity The real number 1 is the multiplicative identity for real numbers: for any real number a, we have a • 1 = 1 • a = a In this section, we define a multiplicative identity matrix I that has properties similar to those of the number 1. We use the definition of this matrix I to find the multiplicative inverse of any square matrix that has an inverse.

Identity Matrix If I is to be the identity matrix, both of the products AI and IA must equal A. The identity matrix only exists for square matrices.

Examples of Identity Matrices

Determining if Matrices are Inverses of Each Other Recall that a number multiplied by its multiplicative inverse yields a product of 1. Similarly, the product of matrix A and its multiplicative inverse matrix A (read “A-inverse”) is I, the identity matrix. So, to prove that two matrices are inverses of each other, show that their product, regardless of the order they’re multiplied, is always the identity matrix.

Example 1 Prove or disprove that the matrices below are inverses of each other. a.) b.) c.)

Finding the Inverse of a Matrix

Row Operations on Matrices

Example 2 Find the inverse, if it exists, for each matrix. a.) b.) c.)

Shortcut for Finding the Inverse of a 2 x 2 Matrix If a matrix is of the form then the inverse can be found by calculating: Note: ad – bc ≠ 0.

Example 3 Find the inverse of the matrix below using the shortcut method.

Solution to Example 3 To find the inverse of the matrix use the formula and simplify.

Solution to Example 3 (continued)