Page 22 - A Course in Linear Algebra with Applications

6 Chapter One: Matrix Algebra

3. Using the fact that matrices have a rectangular shape, say
how many different zero matrices can be formed using a total
of 12 zeros.
4. For every integer n > 1 there are always at least two zero
matrices that can be formed using a total of n zeros. For
which n are there exactly two such zero matrices?

5. Which matrices are both upper and lower triangular?

1.2 Operations with Matrices

We shall now introduce a number of standard operations
that can be performed on matrices, among them addition,
scalar multiplication and multiplication. We shall then de-
scribe the principal properties of these operations. Our object
in so doing is to develop a systematic means of performing cal-
culations with matrices.

(i) Addition and subtraction
Let A and B be two mxn matrices; as usual write a^ and bij
for their respective (i,j) entries. Define the sum A + B to be
the mxn matrix whose (i,j) entry is a^ + b^; thus to form
the matrix A + B we simply add corresponding entries of A
and B. Similarly, the difference A — B is the mxn matrix
whose (i,j) entry is a^- — b^. However A + B and A — B
are not defined if A and B do not have the same numbers of
rows and columns.
(ii) Scalar multiplication

By a scalar we shall mean a number, as opposed to a matrix
or array of numbers. Let c be a scalar and A an mxn matrix.
The scalar multiple cA is the mxn matrix whose (i, j) entry
is caij. Thus to form cA we multiply every entry of A by the
(
scalar c. The matrix -l)A is usually written -A; it is called
the negative of A since it has the property that A + (-A) = 0.