Question

Find a non-zero, two-by-two matrix such that:
[-6, -5] x [__, __] = [0, 0]
[24, 20] [__, __] [0, 0]

These are all 2x2 matrices.
How do we find the missing numbers?

Answer 1

Not true.
\[\left[\begin{matrix}-5 & -5 \\ 6 & 6\end{matrix}\right]\]
will do the trick.

Answer 2

Wondering how I obtained the above answer? Simply carry out the following matrix multiplication:
\[\left[\begin{matrix}-6 & -5 \\ 24 & 20\end{matrix}\right]\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]= \left[\begin{matrix}0 & 0 \\ 0 & 0\end{matrix}\right]\]
You'll get two systems of two equations in two unknowns. The first two will involve a and c the second b and d. They are homogeneous equations (right hand sides are zero), but they are dependent (one equation in a pair is a multiple of another.) So we can let one of the variables be a parameter and solve for the other. There are an infinite number of solutions. Any multiple of the matrix I gave will also work.