1. Inverse of A 2. Inverse of a 2x2 Matrix 3. Matrix With No Inverse 4. Solving a Matrix Equation 1

 The inverse of a square matrix A, denoted by A - 1, is a square matrix with the property  A -1 A = AA -1 = I,  where I is an identity matrix of the same size. 2

 Verify that is the inverse of 3 checks

4

 Find the inverse of 5 1. Interchange: 2. Change signs: 3. Divide:

 A matrix has no inverse if 6

7 has no inverse. has an inverse.

 Solving a Matrix Equation If the matrix A has an inverse, then the solution of the matrix equation  AX = B is given by X = A -1 B. 8

 Use a matrix equation to solve 9 The matrix form of the equation is

 The inverse of a square matrix A is a square matrix A -1 with property that A -1 A = I and AA -1 = I, where I is the identity matrix. 10  A 2x2 matrix has an inverse if = ad - bc ≠ 0. If so, the inverse matrix is

 A system of linear equations can be written in the form AX = B, where A is a rectangular matrix of coefficients of the variables, X is a column of variables, and B is a column matrix of the constants from the right side of the system. If the matrix A has an inverse, then the solution of the equation is given by X = A -1 B. 11