How do I turn a 2x3 matrix into a 3x3 matrix

Question

I'm sorry if this is a stupid question but I just can't find the answer I need. I have the following matrix:-

 A  |6 6 0|
    |9 0 0|

Each column represents co-ordinates on a grid. Now to find the inverse of "A" I need to create this into a 3x3 square matrix, to do this I add 001 as the 3rd row...

 B  |6 6 0|
    |9 0 0|
    |0 0 1|

I do this simply because it is what I have seen in the online examples.

My question is, what is the method to calculate/add the 3rd row of a 2x3 matrix in this situation?

Answer 1

It is not possible to take the inverse of a matrix that is not squared.. I assume that would like to just extend the matrix in order to make i squared, the reason why you use the [0 0 1] is to make the matrix consistent..

Actually you matrix represent two equations with three variables..

A:

    6*x_1 + 6*x_2 + 0*x_3 = 0
    9*x_1 + 0*x_2 + 0*x_3 = 0

this is not consistent but by adding the last row you get

B:

    6*x_1 + 6*x_2 + 0*x_3 = 0
    9*x_1 + 0*x_2 + 0*x_3 = 0
    0*x_1 + 0*x_2 + 1*x_3 = 0

this matrix exists on echelon form

[1 0 0]
[0 1 0]
[0 0 1]

so by adding the last row you are not changing the matrix

you would properly get same result just by reduce it to

[6 6]
[9 0]

Answer 2

Here is a simple way to do it:

s = size(A);
B = eye(max(s));
B(1:s(1),1:s(2)) = A