Determining invertibility of a matrix

[allanmulin]

Let A, B, C, D be matrices such that:

AB + CD = 0

and

B is invertible. Moreover, consider the dimension restrictions:

A(m x n), B(n x n), C(m x m), D(m x n)

If C is a square matrix, is there a way to show that it is also invertible with only the above conditions?

 

[AlephZero]

If you take A = D = 0, then AB + CD = 0 for any matrix C, so you can't prove C is invertible.

 

[allanmulin]

A and D are non-zero matrices, forget to say.

 

[micromass]

Take

$$A=\left(\begin{array}{cc} 1 & 1\\ 1 & 1 \end{array}\right), B=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right), C=\left(\begin{array}{cc} 1 & 1\\ 1 & 1 \end{array}\right), D=\left(\begin{array}{cc} -1 & 0\\ 0 & -1 \end{array}\right)$$

 

[allanmulin]

A and D are rectangular, not square.

 

[micromass]

Take

$$A=\left(\begin{array}{c} 1\\ 1\end{array}\right), B=\left(\begin{array}{c} 1 \end{array}\right), C=\left(\begin{array}{cc} 1 & 1\\ 1 & 1 \end{array}\right), D=\left(\begin{array}{c} -1\\ 0 \end{array}\right)$$

 

[allanmulin]

The example you gave yields incompatible dimensions: AB is 1x2 and CD is 2x1.

 

[micromass]

The example you gave have yields incompatible dimensions: AB is 1x2 and CD is 2x1.

I was editing. Check again.

You can really find these things for yourself.

 

[allanmulin]

That's what I am trying to do! B and C should have the same dimension.

A(m x n), B(n x n), C(m x m), D(m x n)

 

[micromass]

That's what I am trying to do! B and C should have the same dimension.

A(m x n), B(n x n), C(m x m), D(m x n)

Here, m=2 and n=1.

 

[allanmulin]

Yeah, you're right. Thanks a lot.

It seems there is some more information in my physical problem to show that C should be invertible, but could not find it yet.