Mathematics 243–Algebraic Structures, section 1 Discussion 2 – Matrices, and Matrix Sums and Products September 19, 2003

Background

Today, we will introduce a new class of mathematical objects called matrices and sum and product operations on them. This will give us some important additional examples of binary operations to look at. (And we will also use matrices extensively next semester in Linear Algebra.) In general, a matrix is a rectangular array of numbers. There can be any number of rows (horizontal) and columns (vertical) in a matrix. But to begin with simple cases, we will only consider matrices with at most 2 rows or columns. For instance in the following equations

A = ( \begin{matrix}1 & −8 \\ 7 & 6 \end{matrix} ) ,  B = ( 12 −1 ) ,  C = ( \begin{matrix}1 \\ −2 \end{matrix} ) ,

A has 2 rows and 2 columns (we also say A is a 2 × 2 matrix), B has 1 row and 2 columns, and C has 2 rows and 1 column.

Let M_{mn} be the set of all m × n matrices (with real entries). We can define a binary sum operation on M_{mn} by adding the corresponding entries in the matrices. For instance with m = n = 2:

( \begin{matrix}1 & −8 \\ 7 & 6 \end{matrix} ) + ( \begin{matrix}0 & 2 \\ 1 & 2 \end{matrix} ) = ( \begin{matrix}1 + 0 & −8 + 2 \\ 7 + 1 & 6 + 2 \end{matrix} ) = ( \begin{matrix}1 & −6 \\ 8 & 8 \end{matrix} )

On the other hand, if two matrices have different sizes they just can’t be added – the sum is not defined in that case.

Two matrices A, B can be multiplied to produce another matrix C = AB if the number of columns of A is the same as the number of rows of B. The product then has the same number of rows as A and the same number of columns as B. The entries of the product matrix can be formed by taking dot products of the rows of A with the columns of B as in the following example:

(1)  \begin{aligned}( \begin{matrix}1 & −8 \\ 7 & 6 \end{matrix} ) · ( \begin{matrix}0 & 2 \\ 1 & 2 \end{matrix} ) & = ( \begin{matrix}1 · 0 + −8 · 1 & 1 · 2 + −8 · 2 \\ 7 · 0 + 6 · 1 & 7 · 2 + 6 · 2 \end{matrix} ) \\ & = ( \begin{matrix}−8 & −14 \\ 6 & 26 \end{matrix} ) \end{aligned}

This product defines a binary operation on the set M_{22} of all 2 × 2 matrices.

Discussion Questions

A) Show that the sum operation on M_{12} or M_{21} is essentially the same as the vector sum operation on R^{2} you have studied in MATH 241.

B)

1) Is the matrix sum operation on M_{22} commutative? Prove or give a counterexample.

2) Is it associative? Prove or give a counterexample.

3) Is there an identity element? If so, what matrix is it? If not, say why not.

4) Does every element have an inverse under this operation? Prove or give a counterexample.

C) 1) Compute the matrix product

( \begin{matrix}0 & 2 \\ 1 & 2 \end{matrix} ) · ( \begin{matrix}1 & −8 \\ 7 & 6 \end{matrix} )

(same matrices as in (1) above, but in the opposite order). What does this say about the product operation on M_{22}?

2) Is the product on M_{22} associative? Prove or give a counterexample.

3) Show that the matrix

I = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )

is an identity element for matrix multiplication in M_{22}.

4) Does the matrix A = ( \begin{matrix}0 & 2 \\ 1 & 2 \end{matrix} ) have a multiplicative inverse? Try to find one by

solving for the entries in B = ( \begin{matrix}a & b \\ c & d \end{matrix} ) to make AB = I = BA.

Assignment

Group write-ups due in class on Wednesday, September 24.