As always, there you are with two equations staring you dead in the face.

For example:

*x* + 4*y* = 6

and

2*x* + 3*y* = 9

Remember that variables are the letters, and coefficients are just the numbers in front of the variables. Since you foolishly swallowed the red pill, you need to go ahead and plug them into some matrices (plural for matrix). As usual, there are actually practical reasons to do that too, reasons that pop up in real life. More on that later.

Now, image playing a game of Connect Four. In Connect Four, we take a disk-shaped game piece and drop it into the vertical game board. Then the game pieces stack up on top of one another. Building a matrix is similar in that we stack the equations on top of each other within the matrix brackets.

If we were to take the above equations and place them into the matrix, they'd look like this:

That's all a matrix really is—a grid of the numbers and variables that make up equations. Because matrices are so stylish, they wear brackets.

Entries are the numbers or variables inside the brackets.

The rows of a matrix are the horizontal number groups, so in this matrix

row one = 2, -4, 1

and

row two = 3, 3, -5

The columns of a matrix are the vertical number groups, so in this matrix the columns are:

column one = 2, 3,

column two = -4, 3,

and

column three = 1, -5

Sometimes we have to identify entries in matrices in ways that designate where they are. Take this example again:

This is a 2 × 3 matrix, since we put the vertical size first. This is also how they name the coordinates of each entry in the matrix. Any given entry (let's call an entry *e* for now) has a location inside the matrix. For every entry it has a row location (*r*) and column location (*c*). Then each entry is noted like this:

*e*_{rc}

therefore

2 is entry *e*_{11}, -4 is entry *e*_{12}, 1 is entry *e*_{13}, the first 3 is entry *e*_{21}, the second 3 is entry *e*_{22}, and -5 is entry *e*_{23}.

A **square matrix** has the same number of rows and columns. (They don't get invited to many parties.) Here's an example:

this:

or this (any size goes, except for one lone digit in brackets):

Zero matrices can have any shape or size. They're just matrices full o' zeros. (And you thought the squares were the life of the party.)

Here's an example:

This one is a square zero matrix:

You may think zero matrices seem silly, but they're necessary to do arithmetic with matrices, namely addition and subtraction. Trust us, they're not as useless as they seem. Speaking strangely useful matrices, identity matrices are up next. We need these in order to multiply matrices, divide matrices, and to find the meaning of life. Don't worry; this is simple, too. (Well, two of the three things are simple…the meaning of life might take some time to figure out.)

Identity matrices are like zero matrices, but with 1's added in. Identity matrices also have extra rules. They *have to* do these three things to be identity matrices: (1) they must be square, (2) the diagonal line from the upper left to the lower right must have all entries equal to 1, and (3) the rest of the entries must be 0. Check it:

In multiplication, identity matrices have the same effect on matrices that ones have on numbers. After multiplying by the identity, the resulting matrix (or number) just stays the same.

Example 1

Using this matrix, name the entries and state which entries make up the rows and columns: |

Example 2

What is the difference between an identity matrix and a zero matrix? |

Exercise 1

Express these equations using matrices.

3*x* + *y* = 0

-*x* – 5*y* = 4

Exercise 2

Express these equations using matrices.

3*x* – 4*y* = -7

*x* + *y* = 9

Exercise 3

Express these equations using matrices.

2*x* + 2*y* = 11

6*x* – *y* = 3

Exercise 4

For the matrix below, identify: a) the rows, b) the columns, and c) each entry.

Exercise 5

For the matrix below, identify: a) the rows, b) the columns, and c) each entry.

Exercise 6

For the matrix below, identify: a) the rows, b) the columns, and c) each entry.

Exercise 7

For the matrix below, identify: a) the rows, b) the columns, and c) each entry.

Exercise 8

For the matrix below, identify: a) the rows, b) the columns, and c) each entry.