# How do you find the inverse of A=((9, 6, 12), (6, 4, 8), (12, 8, 16)) ?

Feb 13, 2016

This matrix has no inverse since its determinant is zero.

#### Explanation:

Notice that the third row is twice the second row, so these rows are not linearly independent. So there can be no inverse.

Still need convincing?

$\left\mid \begin{matrix}9 & 6 & 12 \\ 6 & 4 & 8 \\ 12 & 8 & 16\end{matrix} \right\mid$

$= 9 \left\mid \begin{matrix}4 & 8 \\ 8 & 16\end{matrix} \right\mid + 6 \left\mid \begin{matrix}8 & 6 \\ 16 & 12\end{matrix} \right\mid + 12 \left\mid \begin{matrix}6 & 4 \\ 12 & 8\end{matrix} \right\mid$

$= 9 \left(4 \cdot 16 - 8 \cdot 8\right) + 6 \left(8 \cdot 12 - 6 \cdot 16\right) + 12 \left(6 \cdot 8 - 4 \cdot 12\right)$

$= 0 + 0 + 0 = 0$