# How do you solve equations with matrices?

1. Apr 9, 2012

### 3m0k177y

How do you solve equations that have matrices?
heres an example (its just off the top of my head)

3x+y=z+4 where
x=1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
and y=0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
and a follow up question; does the process change if coordinates are involved in the equation?
if so, how?

2. Apr 9, 2012

### AlephZero

Your equation doesn't make any sense. On the left hand side, if x and y are 1x4 matrices then 3x+y is a 1x4 matrix. That's OK. The left hand side is the matrix 3 0 0 4.

So the right hand side must also be a 1x4 matrix, but can't add a matrix to the scalar "4".

I have no idea what you think the array of numbers after the equation means.

3. Apr 10, 2012

### jppike

I will answer your question assuming that you meant 4 to be 4*E, where E is the identity matrix of GL(4,R) (i.e x=E).

To solve the system you are given you simply reduce it to a simple equation in each entry. So if we denote by Xij, Yij, Zij, Eij the entry in the ith row, jth column of X,Y,Z,E respectively, the equation becomes:

3Xij+Yij=Zij+4Eij

for every i and every j. Rearrange:

Zij=3Xij+Yij-4Eij

Hence Z is the matrix:

-1 0 0 1
0 -1 1 0
0 1 -1 0
1 0 0 -1

Hope that helps