If $A = \begin{bmatrix} 0 & 3 \\ 0 & 0 \end{bmatrix}$ and $f(x) = x + x^2 + x^3 + \ldots + x^{2023}$, then $f(A) + I = $

Let $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix}$ and $B = 7A^{20} - 20A^{7} + 2I$,where $I$ is an identity matrix of order $3 \times 3$. If $B = [b_{ij}]$,then $b_{13}$ is equal to $....$

Consider the matrices $A = \begin{bmatrix} 4 & 6 & -1 \\ 3 & 0 & 2 \\ 1 & -2 & 5 \end{bmatrix}$,$B = \begin{bmatrix} 2 & 4 \\ 0 & 1 \\ -1 & 2 \end{bmatrix}$,and $C = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}$. Which of the following matrix products are defined?
$(i) (AB)^T C$
$(ii) C^T C (AB)^T$
$(iii) C^T AB$
$(iv) A^T AB B^T C$

$G = \left\{ \begin{bmatrix} x & x \\ x & x \end{bmatrix} : x \in \mathbb{R} \setminus \{0\} \right\}$ is a group with respect to matrix multiplication. In this group,the inverse of $\begin{bmatrix} 1/3 & 1/3 \\ 1/3 & 1/3 \end{bmatrix}$ is

If $A = \begin{bmatrix} 4 & 6 & -1 \\ 3 & 0 & 2 \\ 1 & -2 & 5 \end{bmatrix}$,$B = \begin{bmatrix} 2 & 4 \\ 0 & 1 \\ -1 & 2 \end{bmatrix}$,and $C = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}$,then which of the following expressions is not defined?

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