I've been doing lots of matrices practice today and I've come across this one which I'm finding quite tricky.
$$
𝐴 =
\begin{pmatrix}
3 & 5 \\
z & -3
\end{pmatrix}
$$
All I'm told is that $A^2$ is a matrix whose entries are all $0$, and I need to find z.
$$
A^2 =
\begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix}
$$
So I suppose I need to write out the second matrix and then I need to find a way of getting to k. Would I first be right to find the determinant?
So
$$
(A_{11} \times A_{22}) – (A_{12} \times A_{21}) = -9 – 5z
$$
I was wondering how I might bring the second mentioned matrix into my working out, because obviously it is important?
Best Answer
You know that $A^2 = 0$. If $A$ were invertible, you could multiply both sides by the inverse of $A$: $$ A^{-1} A^2 = A = A^{-1} 0 = 0 $$ but you know that $A \ne 0$, so $A$ has no inverse, thus its determinant must vanish. Which helps you to determine $z$.