How would I find a non-zero matrix that fits the following equation?

Question

$$\begin{pmatrix}4\:&\:-5\:\\ \:\:-16\:&\:20\end{pmatrix}.\begin{pmatrix}a&b\\ \:\:\:c&d\end{pmatrix}=\begin{pmatrix}0&0\\ \:\:\:0&0\end{pmatrix}$$

I find this problem particularly confusing as you can not find the inverse of a $2 \times 2$ matrix.

set up the equations and solve the system by ironically using linear algebra — user29418, Sep 24 '17 at 05:30
Check that the original matrix is singular - then find an eigenvector associated with the eigenvalue zero. — Mark Bennet, Sep 24 '17 at 05:56

score 3 · Accepted Answer · answered Sep 24 '17 at 05:34

You could find the inverse of the first $2\times 2$ matrix if it were invertible, but it is not. Therefore, there is some nontrivial linear combination of its columns which is equal to the $0$ vector. Since the second column is just $-5/4$ the first column, we might consider $[1,4/5]^{T}$, and observe that this indeed gives $0$ if we multiply the first matrix by this vector. Then for any nonzero $a$ and any $b$, the matrix $$\begin{bmatrix} a&b\\ (4/5)a&(4/5)b\end{bmatrix}$$ is nonzero and has the desired property.

score 1 · Answer 2 · answered Sep 24 '17 at 05:34

Hint:

note that $$4 (5) -5(4) =0$$

Can you use this information to set $a,b,c,d$ to satisfy the matrix equation?

You might want to solve this problem first. $$\begin{pmatrix}4\:&\:-5\:\\ \:\:-16\:&\:20\end{pmatrix}.\begin{pmatrix}a\\ c\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$$

How would I find a non-zero matrix that fits the following equation?

2 Answers2

Linked