2 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?
3 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).
4 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.
6 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
7 The Multiplicative Identity Give the multiplicative identity for matrix B.This identity matrix is I4.
8 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.
9 The Inverse of a MatrixLet 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.
10 The Inverse of a MatrixTo 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.
15 Inverses – Method 1, cont. So the inverse of A = We can check this by multiplying A x A-1
16 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.
17 Finding the Inverse with a Calculator Find the inverse of each matrix using the calculator.
18 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.
19 DeterminantsEach square matrix can be assigned a real number called the determinant of the matrix. It is denoted by the symbolmeans the determinant of A.
20 DeterminantsThe determinant of a 2 x 2 matrix is found as follows:
21 DeterminantsFind the determinant of the matrix.
22 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.
23 Determinants can be used to find the inverse of a matrix.
24 Determinants can be used to find the inverse of a matrix. is called the adjoint of the original matrix. Notice it isfound by switching the entries on the main diagonal and changing the signs of the entries on the other diagonal.