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BASIC PROPERTIES OF MATRICES
# BASIC PROPERTIES OF MATRICES

A matrix having the same number of rows as columns is called a square matrix,
The matrices below are square:

A matrix containing only one row is called a row matrix. Here are some examples:

[5 8], [6 -9 2], and [-4 0 0 0].

Finally, a matrix of only one column, as in part (d) of Example 1, is a column
matrix.

It is customary to use capital letters to name matrices. Subscript notation is
used to name the elements of a matrix, as follows.

Using this notation, the first row, first column element is denoted a11, the
second row, third column element is denoted a23, and the *i*th row, *j*th
column element is denoted *a*ij.

Two matrices are equal if they are the same size and if each corresponding
element, position by position, is equal. Using this definition, the matrices

are not equal (even though they contain the same elements and are the same
size), since the corresponding elements differ.

**DECIDING WHETHER TWO MATRICES ARE EQUAL**

**Example 3**

From the definition of equality given above, the only way that the statement

can be true is if 2 = x, 1 = y, p = -1, and q = 0.

The statement

can never be true, since the two matrices are different sizes. (One is 2 x 1 and
the other is 3 x 1.)

## ADDING MATRICES

At the beginning of this section we used the matrix

where the columns represent the numbers of cartons of three different types of
vi tamins (A, E, and K, respectively), and the rows represent two different
manufacturers (Dexter and Sullivan, respectively). For example, the element 7
represents 7 cartons of vitamin E pills from Dexter, and so on. Suppose another
shipment from these two suppliers is described by the following matrix.

Here, for example, 8 cartons of vitamin K pills arrived from Sullivan. The
number of cartons of each kind of pill that were received from these two
shipments can be found from these two matrices.

In the first shipment, 2 cartons of vitamin A pills were received from Dexter,and in the second shipment, 3 cartons of vitamin A pills were received.
Altogether, 2 + 3, or 5, cartons of these pills were received. Corresponding elements can be
added to find the total number of cartons of each type of pill received.

The last matrix gives the total number of cartons of each type of pill that were
received. For example, 15 cartons of vitamin K pills were received from Dexter.
Generalizing from this example leads to the following definition.

**MATRIX ADDITION:** The sum of two *m x n* matrices A and B is the *m
x n* matrix A + B
in which each element is the sum of the corresponding elements of A and B.

CAUTION: Only matrices of the same size can be added.

The matrices

are different sizes. Therefore, the sum A + B does not exist.

A matrix containing only zero elements is called a zero matrix. For example, [0
0 0] is the 1 X 3 zero matrix, while

is the 2 X 3 zero matrix. A zero matrix can be written of any size.

In Chapter l the additive inverse of a real number a was defined as the real
number -a such that a + (-a) = 0 and = a + a = 0. Given a matrix A, a matrix -A can be found such that A + (-A) = 0, where 0 is the appropriate zero matrix, and -A + A = 0. For example, if

then the elements of matrix -A are the additive inverses of the corresponding
elements of A. (Remember that each element of A is a real number and thus has
an additive inverse.)

To check, first test that A + (-A) equals 0, the appropriate zero matrix.

Then test that -A +A is also 0. Matrix -A is the additive inverse, or negative,
of matrix A. Every matrix has a unique additive inverse.

##

SUBTRACTING MATRICES

Subtraction of real numbers was defined in Chapter L by saying that a - b = a
+ (-b). The same definition is used for subtraction of matrices.

If A and B are matrices of the same size, then

A - B = A + (-B)

**Find each of the following differences.**

The matrices

are different sizes and cannot be subtracted.

If a matrix A is added to itself, each element in the sum is twice as large as
the corresponding element of A. For example,

In the last expression, the number 2 in front of the matrix is called a scalar
to distinguish it from a matrix. The example above suggests the following
definition of multiplication of a matrix by a scalar.

**MULTIPLICATION OF A MATRIX BY A SCALAR: **The product of a scalar k and a
matrix X is the matrix kX, each of whose elements is k times the corresponding
element of X.

**Find each product.**