Basics

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

For example:

x + 4y = 6

and

2x – 3y = 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, imagine 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: 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 a couple matrices, they'd look like this:

That's all a matrix really is—a grid of numbers and/or variables. Because matrices are so stylish, they wear brackets. Each number or variable inside the brackets is called an entry.

The rows of a matrix are the horizontal number groups. Take a gander at this matrix:

In this example, 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 always put the number of rows first. This is also how we 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. Since every entry has a row location (r) and column location (c), each entry is noted like this:

e_rc

That means 2 is entry e₁₁, -4 is entry e₁₂, -1 is entry e₁₃, the first 3 is entry e₂₁, the second 3 is entry e₂₂, and -5 is entry e₂₃.

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

Here's another:

Here's one more, since we're feeling generous (any size goes, except for one lone digit in brackets):

A zero matrix can have any shape or size. It's just a matrix 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:

Zero matrices may seem kinda goofy and pointless, but they're necessary to do arithmetic with matrices, namely addition and subtraction. Trust us, they're not as useless as they seem. Speaking of 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.

An identity matrix is like a zero matrix, but with 1s added in. Identity matrices also have extra rules. A matrix has to do these three things to qualify as an identity matrix: (1) it 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 the number 1 has on numbers. After multiplying by the identity, the resulting matrix (or number) just stays the same, just like any number times 1 equals itself (9 × 1 = 9, etc.).