Finding the Inverse of a Matrix

Properties of Matrices We have discovered that the commutative property for multiplication does not work for matrix multiplication. Let’s consider some of the other properties of real numbers. Is there a multiplicative identity for matrices? Is there a multiplicative inverse for matrices?

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

Finding the Inverse with a Calculator To find the inverse of a matrix using the calculator, enter the matrix into the calculator and use the x-1 key.

Finding the Inverse with a Calculator Find the inverse of each matrix using the calculator.

Finding the Inverse with a Calculator This error message means that the matrix does not have an inverse. A matrix that does not have an inverse is called an invertible matrix.

Determinants Each square matrix can be assigned a real number called the determinant of the matrix. It is denoted by the symbol . means the determinant of A.

Determinants The determinant of a 2 x 2 matrix is found as follows:

Determinants Find the determinant of the matrix.

Determinants Find the determinant of the matrix. If the determinant of a matrix = 0, the matrix does not have an inverse. Matrix H is invertible.

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.

Finding the determinant of a 3 x 3 matrix

Finding the determinant of a 3x3 matrix. One way to find the determinant of a 3x3 matrix is the formula below.

Find the determinant using the formula

Find the determinant using the formula

Find the determinant using the formula

Find the determinant using the formula