Matrix Inversion: A Caution,
  and a Sample Word Problem (page 2 of 2)

Warning: Not all matrices can be inverted. Recall that the inverse of a regular number is its reciprocal, so 4/3 is the inverse of 3/4, 2 is the inverse of 1/2, and so forth. But there is no 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), some matrices cannot be inverted.

Given a matrix A, the inverse A^–1 (if said inverse matrix in fact exists) can be multiplied on either side of A to get the identity. That is, AA^–1 = A^–1A = I. Keeping in mind the rules for matrix multiplication, this says that A must have the same number of rows and columns; that is, A must be square. (Otherwise, the multiplication wouldn't work.) 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.

Always be careful of the order in which you multiply matrices. 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.

There is only one "word problem" sort of exercise 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 converted letters to numbers, and then 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 C on the left, then I know the encoding equation was:

AC = M

To reverse the encoding, I need to multiply by A^–1 on the left:

A^–1AC = A^–1M
C = A^–1M

This gives me:

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

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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