Question

If A = ${\left[\begin{array}{c}a\\ b\\ -a\end{array}\right]}_{3\times 1}{\left[\begin{array}{ccc}a& b& -a\end{array}\right]}_{1\times 3}$ then find wheather $A^{-1}$ exists or not.

Answer 1

A will be 3 X 3 matrix.
$\therefore A=\left[\begin{array}{ccc}{a}^2& ab& -a^2\\ ba& b^2& ba\\ -a^2& -ab& a^2\end{array}\right]$
Clearly |A| = 0 because $R_1$ and $R_3$ are identical. Therefore A is singular and hence its inverse does not exists.