LESSON 5: Matrix Operations and Matrix Inverses

Terminology:

diagonal matrix, zero matrix, matrix-matrix product, commute, transpose, matrix inverse, invertible, singular and nonsingular, determinant, inner (scalar or dot) product, outer product.

Objectives:

learn how to combine matrices linearly; learn about the the matrix-matrix product and its properties; learn about the matrix transpose and its properties; learn about matrix inverses and how to find them; learn how to use matrix algebra to simplify matrix expressions.

Reading Assignment:

Chapter 2, Sections 2.1-2.2 (pages 97-117).

Lesson Outline

Key Ideas and Discussion:

Matrices can be linearly combined just like vectors can be linearly combined; after all, n-vectors are just $n\times 1$ matrices. However, multiplication of matrices is more complicated. An easy way to multiply two matrices A and B, is to use think of Bas a group of columns and use the the matrix-vector product on each column. The columns of C=AB are just products of A with the respective columns of B. For this to work properly, the number of columns of A must be the same as the number of rows of B. Another way to compute C is element by element; c_ij is the inner product of row i of A with column j of B. An unfortunate fact about the matrix product is that usually $AB\neq BA$ (most matrices do not commute). You are probably very used to using ab = ba when working with scalars a and b, but when you are doing matrix algebra, it is very important to remember that you cannot assume AB=BA. One consequence of this is that the transpose of a product of matrices is the product of the transposes of the individual matrices, but with the order of the terms reversed.

Inverses of matrices can be used to find solutions to linear systems. The scalar equation ax=b has the solution x=b/a; the correct analogue for the solution to a linear system $A{\bf x}={\bf b}$ is ${\bf x}=A^{-1}{\bf b}$. An $n\times n$matrix A is invertible (A^-1, the unique inverse, exists) if and only if $A{\bf x}={\bf b}$ has a unique solution for all ${\bf b}$ in Rⁿ. The inverse of a product of square matrices is the product of the inverses of the terms, with the order of the terms reversed. There is a simple formula for inverses of $2 \times 2$ matrices. For larger matrices, elementary row operations can be used to find inverses. The whole process is equivalent to the simultaneous solution of a set of n linear systems whose right-hand sides are the columns of the $n\times n$ identity matrix I. In practice, you take A and augment it with I. Then reduce. A completely (if possible), performing the same row operations on I. When you are finished, the part of the augmented A that was originally occupied by I, now contains A^-1.

Practice Problems:

2.1.1, 5, 9, 11, 15, 25 (pp. 107-109); 2.2.1, 5, 7, 17, 19, 31, 33 (pp. 117-119).

Assignment