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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 A1 (if said inverse matrix in fact exists) can be multiplied on either side of A to get the identity. That is, AA1 = A1A = 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.

 

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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 B1 on B; that is, you would have to multiply on the right:

    AB = C
    ABB
    1 = CB1

    AI = CB1

    A = CB1

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:
    • A = [[ 1 2 3 ][ 0 1 4 ][ 5 6 0 ]]

    Translate the coded message:

      M = [[ 108 8 26 95 69 3 ][ 79 0 13 95 76 1 ][ 238 40 79 114 60 11 ]]

    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:

      matrix operations

    So the inverse matrix is:

      A^(-1) = [[ -24 18 5 ][ 20 -15 4 ][ -5 4 1 ]]

    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 A1 on the left:

      A1AC = A1M
      C
      = A1M

    This gives me:

A^(-1) M = [[ 20 8 5 0 12 1 ][ 23 0 9 19 0 1 ][ 14 0 1 19 19 0 ]]

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

A

B

C

D

E

F

G

H

I

J

K

L

M

1

2

3

4

5

6

7

8

9

10

11

12

13

  Copyright Elizabeth Stapel 2003-2011 All Rights Reserved

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

14

15

16

17

18

19

20

21

22

23

24

25

26

      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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Cite this article as:

Stapel, Elizabeth. "Matrix Inversion: A Caution, and a Sample Word Problem." Purplemath.
    Available from
http://www.purplemath.com/modules/mtrxinvr2.htm.
    Accessed
 

 



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