Finding the Inverse of a Matrix

The Multiplicative Identity The multiplicative identity for real numbers is the number 1. The property is: If a is a real number, then a x 1 = 1 x a = a. In terms of matrices we need a matrix that can be multiplied by a matrix (A) and give a product which is the same matrix (A).

The Multiplicative Identity This matrix exists and it is called the identity matrix. It is named I and it comes in different sizes. It is a square matrix with all 1’s on the main diagonal and all other entries are 0.

The Multiplicative Identity Multiply AI a11= (-2)(1) + (5)(0) = -2 a12= (-2)(0) + (5)(1) = 5 a21= (4)(1) + (0)(0) = 4 a22= (4)(0) + (0)(1) = 0

The Identity Matrix for Multiplication Let A be a square matrix with n rows and n columns. Let I be a matrix with the same dimensions and with 1’s on the main diagonal and 0’s elsewhere. Then AI = IA = A

The Multiplicative Identity Give the multiplicative identity for matrix B. This identity matrix is I4.

The Multiplicative Inverse For every nonzero real number a, there is a real number 1/a such that a(1/a) = 1. In terms of matrices, the product of a square matrix and its inverse is I.

The Inverse of a Matrix Let A be a square matrix with n rows and n columns. If there is an n x n matrix B such that AB = I and BA = I, then A and B are inverses of one another. The inverse of matrix A is denoted by A-1.

The Inverse of a Matrix To show that matrices are inverses of one another, show that the multiplication of the matrices is commutative and results in the identity matrix. Show that A and B are inverses.

The Inverse of a Matrix and 

The Inverse of a Matrix

Finding the Inverse of a Matrix - Method 1 Use the equation AB = I. Write and solve the equation:

Inverses – Method 1, cont.

Inverses – Method 1, cont. So the inverse of A = We can check this by multiplying A x A-1

Determinants can be used to find the inverse of a matrix.

Determinants can be used to find the inverse of a matrix. is called the adjoint of the original matrix. Notice it is found by switching the entries on the main diagonal and changing the signs of the entries on the other diagonal.

Find the multiplicative inverse of:

We can check to see if we are correct by multiplying We can check to see if we are correct by multiplying. Remember that AA-1 = I

Find the inverse using determinants.

Find the inverse No inverse Recall that when the determinant of a matrix is 0 the matrix will not have an inverse because division by 0 is undefined.