Unit 2
Matrix Arithmetic

In Unit 1, you learned how to solve systems of linear equations by reducing the augmented matrix of the system to row-echelon or reduced row-echelon form. Matrices are a powerful tool for studying systems of linear equations, and in this unit, we introduce the rules of matrix arithmetic. If numbers are thought of as 1 × 1 matrices, then matrices in general can be thought of as generalizations of real numbers. [Matrices can have complex number entries, and even vector entries, but we will not consider such situations in this course.] It will turn out, however, that there are important differences between the arithmetic of real numbers and matrix arithmetic.

Objectives

When you have completed this unit you should be able to

1. state, from memory, the definition of “m × n matrix.”
2. determine whether two given matrices are equal.
3. compute the product of a matrix multiplied by a scalar (i.e., a real number).
4. compute the sum of two matrices of the same size.
5. determine whether two given matrices can be multiplied.
6. determine the product of two matrices that can be multiplied.
7. state, from memory, the definition of “the identity matrix.”
8. explain what we mean when we say that a square matrix is “invertible.”
9. compute the inverse of a square matrix.
10. given a polynomial and a square matrix, construct the corresponding matrix polynomial.
11. prove some simple theorems of matrix arithmetic.
12. state, from memory, the definition of the “transpose” of a matrix together with the basic properties of the transpose.
13. state, from memory, the definition of “symmetric matrix” and “anti-symmetric” (or “skew-symmetric”) matrix.
14. write any square matrix as the sum of a symmetric and an anti-symmetric matrix.