## 3.1 Matrix Addition and Scalar Multiplication

(This topic is also in Section 3.1 in Finite Mathematics and Finite Mathematics & Applied Calculus)

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A matrix is just a rectangular array ("grid") of numbers. Here are a few.

 1 0 5 0 0 1
,
 -4 0 5 -1 0 1 1 5 -3 0 0 1
,
 1 -4 12
,     [2   -10   -1] ,
 1 1 5 - 7 0 1 1 0 0 0 0 1 0 0 0 0
To specify the size of a matrix, we need to talk about rows and columns:
 1   5   -7 1   1   0 0   0   1 0   0   0
 Rows go across

 1 1 0 0
 5 1 0 0
 -7 0 1 0
 Columns go down

The above matrix has 4 rows and 3 columns. We refer to it as a 43 matrix. Here is the general definition from Finite Mathematics.

Matrix, Dimension, Entries

An mn matrix A is a rectangular array of real numbers with m rows and n columns. We refer to m and n as the dimensions of the matrix A. The numbers that appear in the matrix are called its entries. We customarily use upper case letters A, B, C, ... for the names of matrices.

Example

 1 0 5 0 0 1
is a matrix

The entries in the first row are 1, 0, and 5. The entries in the second column are 0 and 0.

Referring to the Entries of a Matrix

There is a systematic way of referring to particular entries in a matrix. If i and j are numbers, then the entry in the ith row and jth column of the matrix A is called the ijth entry of A. We usually write this entry as aij or Aij. (If the matrix was called B, we would write its ijth entry as bij or Bij.) Notice that the row number is specified first and the column number second.

Example

LetA=  1 3 5 -1 -8 10 -7 -5 13
.

 a21 = -1 Entry in Row 2 and Column 1 a13 = a31 = a23 = a22 =

Two matrices can be added (or subtracted) if, and only if, they have the same dimensions. (That is, both matrices have matching numbers of rows and columns. For instance, you can't add, say, a 34 matrix to a 44 matrix, but you can add two 34 matrices.)

To add (or subtract) two matrices of the same dimensions, just add (or subtract) the corresponding entries. In other words, if A and B are mn matrices, then A+B and A-B are the mn matrices whose entries are given by

 (A + B)ij = Aij + Bij ijth entry of the sum = sum of the ijth entries (A - B)ij = Aij - Bij ijth entry of the difference = difference of the ijth entries

Examples

 2 -3 1 0 -1 3
+  9 -5 0 13 -1 3
=  11 -8 1 13 -2 6
 2 -3 1 0 -1 3
-  9 -5 0 13 -1 3
=

A matrix all of whose entries are zero is called a zero matrix. Here are some zero matrices.

 0 0 0 0 0 0
,
 0 0 0 0 0 0 0 0 0 0 0 0
,
 0 0 0
,     [0   0   0] ,
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

LetA=
 1 3 5 -1 -8 10 -7 -5 13
and   B=
 0 0 0 0 0 0 0 0 0

Q Then A + B =

Scalar Multiplication

It is traditional when talking about matrices to call individual numbers scalars. For this reason we call the operation of multiplying a matrix by a number scalar multiplication.

In order to motivate scalar multiplication, consider the following.

Q When can a matrix A be added to itself?
A Always, since the expression A+A is the sum of two matrices that certainly have the same dimensions.

Q Can't we write A+A as 2A?
A We certainly can. Notice that, when we compute A+A, we end up doubling every entry in A. Thus, we can think of the expression 2A as telling us to multiply every element in A by 2.

Similarly, 6A is the matrix obtained from A by multiplying each of its entries by 6. More generally, if c is any number, then cA ("c times A") is the matrix obtained from A by multiplying each of its entries by c.

Examples

4  1 3 5 -1 -8 10 -7 -5 13
=  4 12 20 -4 -32 40 -28 -20 52
Each entry multiplied by 4
-3  1 3 5 -1 -8 10 -7 -5 13
=

 0 -1 -1 2
-     2  3 -2 1 0
=

Algebra of Matrices

Addition and scalar multiplcation of matrices satsify rules similar to those for addition and multiplication of real numbers.

Properties of Matrix Addition and Scalar Multiplication

If A, B and C are any mn matrices, and if O is the zero mn matrix, then the following hold.

 A+(B+C) = (A+B)+C Associative law A+B = B+A Commutative law A+O = O+A = Additive identity law A+(-A) = (-A)+A = Additive inverse law c(A+B) = cA+ Distributive law (c+d)A = Distributive law 1A = Scalar unit 0A = Scalar zero

Q What about the product of two matrices?
A That must wait till the next section (press "Next Tutorial" on the side if you're impatient to see matrix multiplication.)

Q OK the above rules are all very nice, but how can we apply them?
A Matrix algebra can be applied in a mind-boggling number of situations. Here is just one.

The following table gives the number of people (in thousands) who visited Australia and South Africa in 1998.*

 FROM             TO Australia South Africa North America 440 190 Europe 950 950 Asia 1,790 200

* Figures are rounded to the nearest 10,000. Sources: South African Dept. of Environmental Affairs and Tourism; Australia Tourist Commission/The New York Times, January 15, 2000, p. C1.

Take A to the 32 matrix whose entries are the 1998 tourism figures.

Q You have predicted that the 2008 figures will be given by 1.2A. This means that, in 2008, tourism figures will have

Q If the matrix B represents the corresponding tourism figures for 2008, then the matrix B-A represents

Q Let D be the matrix
 -40 0 50 100 -300 0

You are told that the tourism figures in 2004 will be given by the matrix A+D. The entry -300 tells you that

Now try some of the exercises in Section 3.1 of Finite Math, or Finite Math and Applied Calc.

Last Updated: April, 2006