# Week 6

## Topics

This is the second of two weeks devoted to solving systems of linear equations. It begins with two sections on matrices

• Chapter 4-4 : Matrix addition, subtraction, multiplication
• Chapter 4-5 : Matrix inverses

after which we cover one more method for solving linear equations:

• Chapter 4-6: The Inverse Matrix method for solving systems of linear equations

We will skip section 4-7.

## Overview of Matrices

An mxn matrix is a rectangular array of mn numbers, arranged in m rows of n numbers each. Matrices can be added, subtracted, and multiplied by numbers or by other matrices, as long as the matrix sizes are compatible.

"Compatible" means different things for different operations: to add or subtract two matrices, they have to be of the same size. To multiply two matrices, their "inner dimensions" have to agree. If A is an mxn matrix, and B is an nxp matrix, and you put the dimensions side by side: (mxn)(nxp), the "inner dimension" is n, and the "outer dimensions" are m and p. The inner dimension disappears, and the result is of size mxp (the outer dimensions).

We use the notation 0 for the zero matrix, which is a matrix full of zeros, and talk about the zero matrix, even though technically there are lots of different zero matrices: a zero matrix of size 2x3, a zero matrix of size 4x2, etc. The zero matrix corresponds to the number 0. For example, A0 = 0A = 0 for any A, A+0 = A, etc.

If the product AB exists, the product BA does not necessarily also exist. Even if it does, it does not have to be of the same size: if A is of size 2x3, and B is of size 3x2, then AB has size 2x2, and BA has size 3x3. Even if A and B are square matrices of the same size, so that both AB and BA are also of the same size, they don't have to be the same: matrix multiplication is not commutative.

Another property of matrix multiplication which is different from multiplying numbers is that it is possible to have two non-zero matrices A, B with AB=0.

Except for the properties mentioned in the last two paragraphs, matrix arithmetic satisfies all the usual properties of standard arithmetic. For example, A(B+C) = AB + AC, A(BC) = (AB)C, and so on.

The identity matrix of size nxn is the matrix I which has 1 on the principal diagonal, 0 everywhere else. It has the property that IA = A and AI = A, so I corresponds to the number 1. As with the zero matrix, there are really lots of different identity matrices, one for each (square) size.

If you can find a pair of matrices A, B with AB=I and BA=I, then B is called the inverse of A, written A-1. Every number except 0 has an inverse (the inverse of 2 is (1/2)), but not every matrix has an inverse. However, if an inverse exists, it is unique.

Inverse matrices can be used to solve matrix equations. If AX=B, then X=A-1B. Pay attention to the order: X=BA-1 is the solution to XA = B, which is not the same.

A more complicated example is this: Solve AX+X=B.

Solution: AX+X = AX+IX = (A+I)X = B, so B=(A+I)-1B.

## Assignments

Read the textbook and do the homework assignment HW 5.

Last Updated: Wednesday, August 5, 2015