| Introduction and Basic Operations |
Matrices, though they may appear weird objects at first, are a very important
tool in expressing and discussing problems which arise from real life cases.
Our first example deals with economics. Indeed, consider two families A and B
(though we may easily take more than two). Every month, the two families have
expenses such as: utilities, health, entertainment, food, etc... Let us restrict
ourselves to: food, utilities, and health. How would one represent the data
collected? Many ways are available but one of them has an advantage of combining
the data so that it is easy to manipulate them. Indeed, we will write the data
as follows:
If we have no problem confusing the names and what the expenses are, then we may
write
This is what we call a Matrix. The size of the matrix, as a block, is
defined by the number of Rows and the number of Columns. In this
case, the above matrix has 2 rows and 3 columns. You may easily come up with a
matrix which has m rows and n columns. In this case, we say that the matrix is a
(mxn) matrix (pronounce m-by-n matrix). Keep in mind that the first entry
(meaning m) is the number of rows while the second entry (n) is the number of
columns. Our above matrix is a (2x3) matrix.
When the numbers of rows and columns are equal, we call the matrix a
square matrix. A square matrix of order n, is a (nxn) matrix.
Back to our example, let us assume, for example, that the matrices for the
months of January, February, and March are
To make sure that the reader knows what these numbers mean, you should be able
to give the Health-expenses for family A and Food-expenses for family B during
the month of February. The answers are 250 and 600. The next question may sound
easy to answer, but requires a new concept in the matrix context. Indeed, what
is the matrix-expense for the two families for the first quarter? The idea is to
add the three matrices above. It is easy to determine the total expenses for
each family and each item, then the answer is
So how do we add matrices? An approach is given by the above example. The answer
is to add entries one by one. For example, we have
Clearly, if you want to double a matrix, it is enough to add the matrix to
itself. So we have
which implies
This suggests the following rule
and for any number
 ,
we will have
Let us summarize these two rules about matrices.
-
- Addition of Matrices: In order to add two matrices, we add the
entries one by one.
Note: Matrices involved in the addition operation must have the same size.
-
- Multiplication of a Matrix by a Number: In order to multiply a
matrix by a number, you multiply every entry by the given number.
Keep in mind that we always write numbers to the left and matrices to the
right (in the case of multiplication).
What about subtracting two matrices? It is easy, since subtraction is a
combination of the two above rules. Indeed, if M and N are two
matrices, then we will write
M-N = M + (-1)N
So first, you multiply the matrix N by -1, and then add the result to the
matrix M.
Example. Consider the three matrices J, F, and M
from above. Evaluate
Answer. We have
and since
we get
To compute J-M, we note first that
Since J-M = J + (-1)M,
we get
And finally, for J-F+2M, we have a choice. Here we would
like to emphasize the fact that addition of matrices may involve more than one
matrix. In this case, you may perform the calculations in any order. This is
called associativity of the operations. So first we will take care of -F
and 2M to get
Since
J-F+2M = J
+ (-1)F + 2M, we get
So first we will evaluate J-F to get
to which we add 2M, to finally obtain
For the addition of matrices, one special matrix plays a role similar to the
number zero. Indeed, if we consider the matrix with all its entries equal to 0,
then it is easy to check that this matrix has behavior similar to the number
zero. For example, we have
and
What about multiplying two matrices? Such operation exists but the
calculations involved are complicated.
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