Matrices 3 1

Matrix Algebra In this lecture we compare matrix addition and multiplication with the addition and multiplication of real numbers. We shall see that there are similarities, but one has to be more careful with matrix operations, as they lack some of the properties we usually take for granted. We look at commutativity, zero matrices, identity matrices, and inverses. 2

Commutativity Matrix addition is commutative: if A + B can be calculated then B + A also exists and A + B = B + A. But multiplication is not commutative: It may be that AB is defined but BA is undefined. (Let A be 2 × 3 and B be 3 × 3.) It may be that AB and BA are both defined, but different sizes. (Let A be 2 × 3 and B be 3 × 2.) 3

Commutativity It may be that AB and BA are both defined, and the same size, but give different matrices. For example: whereas 4

Zero Matrices If x is any real number, then 0+x = x and x+0 = x. A zero matrix is a matrix all of whose entries are zero. So is the 2 × 3 zero matrix, and of course one can have a zero matrix of any size. 5

Zero Matrices If A is any matrix and O is the zero matrix with the same size as A, then O + A = A = A + O For example 6

Identity Matrices If x is any real number, then 1·x = x and x·1 = x. For matrices we have a similar property, provided we restrict ourselves to square matrices, so that multiplication is always defined and delivers a product of the same size. A square matrix is one that has as many rows as columns, and so is n×n. The n×n identity matrix In has 1’s on the diagonal from top left to bottom right, and 0’s elsewhere. 7

Identity Matrices So If A is any n × n matrix then InA = A = AIn For example 8

The Idea Of An Inverse If x is any real number, then x+(-x) = 0 = -x+x. We say that -x is the additive inverse of x. Similarly, for any matrix A, we have A + (-1)A = O = (-1)A + A For example Thus every matrix has an additive inverse. 9

The Idea Of An Inverse If x is any nonzero real number, then We say that (or x-1) is the multiplicative inverse of x. Note that not every real number has a multiplicative inverse, just the nonzero numbers. If we hope to find a similar property for matrices, we should once again restrict ourselves to square matrices, so that multiplication is always defined. Do some matrices have multiplicative inverses? Which ones? And how do we find the multiplicative inverse of a matrix, assuming it has one? 10

Invertible Matrices Let A be any n × n matrix. Then A is invertible iff the rank of A is n. (We shall not prove this fact.) To put it differently, A has a multiplicative inverse A-1 iff no rows consisting just of 0’s are produced when we use elementary row transformations to reduce A to echelon form. 11

Invertible Matrices For example, let A = Then A is invertible, since R2 ⟶ 4R2 - R1 gives which is in echelon form and has no rows consisting entirely of zeros. 12

Finding Inverses Suppose we have a matrix like A = which has an inverse. How do we find the inverse? The most obvious method is to write 13

Finding Inverses and to solve the corresponding 2 systems of linear equations for the unknowns x1, y1 and x2, y2: 2x1 + 2y1 = 1 x1 + y1 = 0 and 2x2 + 2y2 = 0 x2 + y2 = 1 14

The Gauss-Jordan Method The Gauss-Jordan method streamlines the process of finding A-1 for any square matrix A that is invertible. 1. Write down the matrix A and on its right an identity matrix of the same size. 2. Perform elementary row-transformations on the lefthand matrix (which starts off as A) so as to change it into an identity matrix, and at the same time perform the same elementary transformations on the righthand matrix (which starts of as the identity matrix). 15

The Gauss-Jordan Method When the lefthand matrix has changed into the identity matrix, the righthand matrix will be the inverse of A. If A is not invertible, then the lefthand matrix will not change into an identity matrix but instead will have a row consisting entirely of zeros. 16

Example We find A-1 for A = Step 1: Step 2: Now R1 ⟶ R1, followed by R2 ⟶ 2 R2 - R1, and finishing with R1 ⟶ R1 - R2 give 17

Example so that A-1 = We can check this by multiplying: 18

What Use Are Inverse Matrices? Consider the system of linear equations 2x + 2y = 4 x + y = 3 which can be written as a matrix equation 19

What Use Are Inverse Matrices? If A-1 exists, then a matrix equation AX = B has the solution X = A-1 B, since we may multiply from the left by A-1 on both sides of AX = B to get A-1 AX = A-1 B, and A-1 AX = IX = X. In the case of our example we know A-1 and so so that x = -2 and y = 4. 20