Subject: Linear Algebra

Name: Hoda

Who is asking: Student
Level: All


I have a question about matrix multiplication; well, actually, matrix division. I am assuming that matrices are divided the same as they are multiplied; that is, row by column. But what happens if you have to divide by zero? How does this affect the resulting matrix?

Specifically, the problem I am referring to:

To solve for A, you divide the right matrix by the one on the left.

I come up with the following:

What do I do with that 6/0?

Hi Hoda,

Unfortunately the matrix "division" operation you defined doesn't work the way you want it to.

Compare your problem to a problem with numbers:

6x = 3 To solve for x=1/2 you could divide both sides by 6, or you could multiply both sides by (1/6). If you do the latter, you obtain

(1/6)(6) x = (1/6)(3). Note that (1/6)(6)=1, and 1 times x is x. Hence we have

x = (1/6)(3) = 1/2

The point of this very basic example is to illustrate the same technique for matrices. In matrix multiplication the identity matrix I palys the roll of 1 since, for any 2 by 2 matrix A, I x A = A.

Note that if we multiply

by the matrix on the left side of the equation,

we get

the identity matrix I

compare to

(1/6) (6) = 1

Hence your system by pre-multiplying both sides by the above matrix. You get:

which reduces to your answer,

For more information on such matrix inverses, and how to find them (it looks like we "pulled it out of a hat") consult a Linear Algebra text.

Hope this gives you a start,
Patrick and Penny

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