Question

Is it possible to find an inverse of a non-square matrix? I got a problem from a professor with Matrix A = 4x5 matrix. The Matrix A multiplied by B gives a matrix 4x5. I'm given the Matrix A
[1 -1 2 5 4
3 -2 1 4 2
0 1 2 -1 3
2 1 5 0 1]

AB is
[2 1 6 5 -8
6 2 4 4 -4
0 -1 6 -1 -6
4 -1 15 0 -2]

Solve for B? If I can't find an inverse, can I find a solution for B?

Answer 1

Here is a nice trick:

If you multiply \[A^{T}\] on the left of each side of your equality, you will have:

\[A^{T}\times A \times B = A^{T}\times AB\] Well, if A is a 4x5 matrix, then \[A^{T}\] will be a 5x4 matrix, furthermore, the product \[A^{T}\times A \] will be a 5x5 matrix, is that ok? We shall call \[A^{T}\times A \ = \Omega\] Now, we can rewrite our initial problem as:
\[\Omega \times B = A ^{T}\times AB\] If you remember, this "Omega" matrix is the 5x5 square matrix we obtained only by multiplying A by it's transposed, so we know how it looks like and it means we can get it's inverse, after we get it, multiply it on both sides of the equation:
\[\ B = I_{5} \times B = \Omega ^{-1}\times \Omega \times B = \Omega ^{-1}\times A ^{T}\times AB\]Where "i-five" is the 5x5 identity matrix. Now you can easily calculate it using matlab, since you got all the matrices on the right side.

My answer was

8.2188 0.3281 20.1758 3.5195 -11.7969
16.7813 5.0508 19.4258 13.2344 -18.4297
-0.2500 -0.6875 1.8438 -1.3438 4.5000
-0.8438 0.6719 -4.9727 0.8711 0.2344
-0.0208 0.0186 -0.0391 -0.0034 -1.9796

Altough I can be wrong. Hope it helped and I'm sorry for any mess I made.

Answer 2

This answer has an important mistake. The 5 x 5 matrix Ohmega has rank four, so it is not invertible. You could use the Moore Penrose generalized inverse as a pseudo-inverse. Then the solution would be A^+(AB) = B. For your numbers, A^+ is

-0.2899 0.3750 0.1319 0.0122
0.1016 -0.1821 -0.0369 0.0643
0.0988 -0.1343 -0.1124 0.2119
0.2515 -0.1404 -0.2568 0.0441
-0.0159 0.1034 0.3350 -0.1480

Then Bhat= A^+AB is the best guess for B. It is

1.7188 0.3160 0.7344 -0.0816 0.0035
-0.6319 -0.2900 0.6254 -0.1833 0.0078
0.2396 -0.2692 2.5596 0.0695 -0.0030
-0.1632 0.1833 0.0681 0.9527 0.0020
-0.0035 0.0039 0.1078 -0.0010 -2.0000

For sure ABhat = AA^+AB = AB.

Answer 3

Perhaps I'm being simplistic here, but it seems that all of the column vectors in AB are scalar multiples of the column vectors in A except for the entry in row 2 column 3 of AB. If that entry was a 3, the answer for B would be

2 0 0 0 0
0 -1 0 0 0
0 0 3 0 0
0 0 0 1 0
0 0 0 0 -2

Is it possible that there was a copying mistake?

Answer 4

It is not possible to inverse the non-square matrix.

To inverse matrix the two candition have to accomplished
- if u want to inverse A in K - body and n - dimension -
1) \[A,B \in M _{n \times n}\left( K \right)\]
2) \[A \times B = I\]