Q.If $A = \begin{bmatrix}1 & -2 & 3\\1 & 2 & 1\\x & 2 & -3\end{bmatrix}$ is singular, find the value of $x$.

A square matrix $A$ is called:

• Singular if $|A|=0$;
• Non-singular if $|A|\ne 0$.

This single number test decides an enormous amount about a square matrix's behaviour (for instance, whether it has a multiplicative inverse — though inverses are developed in a later course, the singular/non-singular label is introduced here purely from the determinant).

Key fact. If $A$ and $B$ are non-singular matrices of the same order, then $AB$ and $BA$ are also non-singular, because $|AB| = |A||B| = |BA|$ is a product of two nonzero numbers, hence itself nonzero. …