Properties of Matrices (Edexcel A Level Further Maths: Core Pure): Exam Questions

$\mathbf{M}=\left(\begin{array}{ccc}k& 5& 7\\ 1& 1& 1\\ 2& 1& -1\end{array}\right)$ where $k$ is a constant

Given that $k\ne 4,$ find, in terms of $k$, the inverse of the matrix $\mathbf{M}$.

$\mathbf{M}=\left(\begin{array}{ccc}2& a& 4\\ 1& -1& -1\\ -1& 2& -1\end{array}\right)$

where $a$ is a constant.

For which values of $a$ does the matrix $\mathbf{M}$ have an inverse?

Given that $\mathbf{M}$ is non-singular, find ${\mathbf{M}}^{-1}$ in terms of $a$

$\mathbf{A}=\left(\begin{array}{ccc}k& 1& -1\\ 1& 2& 1\\ 1& 1& 2\end{array}\right)$

Given that $\mathbf{A}$ is non-singular, find, in terms of $k$, ${\mathbf{A}}^{-1}$.

Determine, in simplest form in terms of $k$, the coordinates of the point where the following planes intersect.

$kx+y-z=6$

$x+2y+z=3$

$x+y+2z=0$