If the matrix product $AB = O$,where $O$ is the null matrix,then which of the following is necessarily true?

If $A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}$,then for what value of $\lambda$ is $A^2 = O$?

If $I$ is the identity matrix of order $2$ and $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then for $n \geq 1$, mathematical induction gives:

If $A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$,then show that $|2A| = 4|A|$.

The order of the matrix product $\left[ {x\,y\,z} \right]\,\left[ {\begin{array}{*{20}{c}} a&h&g\\ h&b&f\\ g&f&c \end{array}} \right]\,\left[ {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right]$ is:

Show that $\left[ {\begin{array}{cc} 5 & -1 \\ 6 & 7 \end{array}} \right] \left[ {\begin{array}{cc} 2 & 1 \\ 3 & 4 \end{array}} \right] \ne \left[ {\begin{array}{cc} 2 & 1 \\ 3 & 4 \end{array}} \right] \left[ {\begin{array}{cc} 5 & -1 \\ 6 & 7 \end{array}} \right]$