Warning: Not all matrices can be inverted. In fact, most matrices can *not* be inverted. The reasons why not are similar to why the number zero has no reciprocal.

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Recall that the multiplicative inverse of a regular number is its reciprocal, so is the inverse of , 2 is the inverse of , and so forth.

But there is no multiplicative inverse for 0, because you cannot flip 0/1 to get 1/0 (since division by zero doesn't work). For similar reasons (which you may or may not encounter in later studies), most matrices cannot be inverted.

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Given an invertible matrix *A*, the inverse *A*^{−1} can be multiplied on either side of *A* to get the identity. That is, *AA*^{−1} = *A*^{−1}*A* = *I*. However, given a particular matrix equation, the matrix for which you are solving will tell you on which side you need to multiply the inverse.

For instance, if you are given *B* and *C* and asked to solve the matrix equation *AB* = *C* for *A*, you would need to cancel off *B*. To do this, you would have to multiply *B*^{−1} on *B*; that is, you would have to multiply *on the right*:

*AB* = *C*

*ABB*^{−1} = *CB*^{−1}

*AI* = *CB*^{−1}

*A* = *CB*^{−1}

The side on which you multiply will depend upon the exercise. Take the time to get this right.

Keeping in mind the rules for matrix multiplication, the defining equation of inverse matrices, *AA*^{−1} = *A*^{−1}*A* = *I*, says that *A* must have the same number of rows as it has columns; that is, *A* must be square. Otherwise, the multiplication wouldn't work. So, if the matrix isn't square, it cannot have a (properly two-sided) inverse. However, while all invertible matrices are square, not all square matrices are invertible.

This fact can be turned into a trick question by, say, giving you a matrix containing at least some variables (so you can't work numerically with your calculator), or of such size that it's unworkable, like this:

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You'll be sitting there, starting at it, wondering what to do. Then you notice the indices (that is, the index numbers that are subscripted on the various entries), and realize that, since 999 > 501, this matrix has more columns than it has rows. This means that the matrix isn't square, so it can't possibly be invertible. And that's your answer: "No, it's not invertable."

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Inverse matrices are used to solve matrix equations. These equations are usually derived from systems of linear equations.

There is only one type of word problem that I can think of that uses matrices and their inverses, and it involves coding and decoding.

- You receive a coded message. You know that each letter of the original message was replaced with a one- or two-digit number corresponding to its placement in the English alphabet, so E is represented by 5 and W by 23; spaces in the message are indicated by zeroes. You also know that the message was transformed (encoded) by multiplying the message on the left by the following matrix:

Translate the coded message:

To do the decoding, I have to undo the matrix multiplication. To undo the multiplication, I need to multiply by the inverse of the encoding matrix. So my first step is to invert the coding matrix:

So the inverse matrix is:

My correspondent had converted letters to numbers, and then had entered those numbers into a matrix *C*. He then multiplied by this matrix by the encoding matrix *A*, and sent me the message matrix *M*. Since the encoding was done by multiplying *A* with *C* by putting *A* *on the left* of *C*, then I know the encoding equation was:

*AC* = *M*

To reverse the encoding, I need to multiply by *A*^{−1} on the left, so that the inverse matrix will cancel off the encoding matrix:

*A*^{−1}*AC* = *A*^{−1}*M*
*C* = *A*^{−1}*M*

This gives me:

At this point, the solution is a simple matter of doing the number-to-letter correspondence:

A ⇒ 1 | B ⇒ 2 | C ⇒ 3 | D ⇒ 4

E ⇒ 5 | F ⇒ 6 | G ⇒ 7 | H ⇒ 8

I ⇒ 9 | J ⇒ 10 | K ⇒ 11 | L ⇒ 12

M ⇒ 13 | N ⇒ 14 | O ⇒ 15 | P ⇒ 16

Q ⇒ 17 | R ⇒ 18 | S ⇒ 19 | T ⇒ 20

U ⇒ 21 | V ⇒ 22 | W ⇒ 23 | X ⇒ 24

Y ⇒ 25 | Z ⇒ 26

The message begins as:

T H E L A W I S A N A ...

(You can complete the decoding to view the original quotation.)

A better code could be constructed by shifting the letters first, adding some value to each letter's coded result, using a larger invertible matrix, etc, etc. The above example is fairly simplistic, and is intended only to show you the general methodology.

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