Section 9.5 Inverses of Matrices Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Objectives Find the inverse of a square matrix, if it exists. Use inverses of matrices to solve systems of equations.

The Identity Matrix

Example For find each of the following. a) AI b) IA

Inverse of a Matrix For an n  n matrix A, if there is a matrix A1 for which A1 • A = I = A • A1, then A1 is the inverse of A. Verify that is the inverse of . We show that BA = I = AB.

Finding an Inverse Matrix To find an inverse, we first form an augmented matrix consisting of A on the left side and the identity matrix on the right side. Then we attempt to transform the augmented matrix to one of the form The 2  2 identity matrix matrix A

Example Find A1, where A = .

Example continued Thus, A1 =

Notes If a matrix has an inverse, we say that it is invertible, or nonsingular. When we cannot obtain the identity matrix on the left using the Gauss-Jordan method, then no inverse exists.

Solving Systems of Equations Matrix Solutions of Systems of Equations For a system of n linear equations in n variables, AX = B, if A is an invertible matrix, then the unique solution of the system is given by X = A1B. Since matrix multiplication is not commutative in general, care must be taken to multiply on the left by A1.

Example Use an inverse matrix to solve the following system of equations: 3x + 4y = 5 5x + 7y = 9 We write an equivalent matrix, AX = B: In the previous example we found A1 =

Example continued We now have X = A1B. The solution of the system of equations is (1, 2).