Practice set second

1).  If  $A=\left[\begin{array}{rr}3 & 1 \\-1 & 2\end{array}\right] \;\;\text {Find }\;\; A^2-5 A+71$.

2). Given that  $2 X+3 Y=\left[\begin{array}{ll}2 & 3 \\4 & 0\end{array}\right]$  and  $3 X+2 Y=\left[\begin{array}{rr}2 & -2 \\-1 & 5\end{array}\right]$, find the value  of $X$ and $Y$.

3). Given that  $A=\begin{bmatrix} 3 & -2 \\ 4 &-2\end{bmatrix}$  and  $I=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ , find the value of $k$ if  $A^{2}=kA-2I$.

4). Given that $A=\begin{bmatrix} -2 \\ 4 \\ 5 \end{bmatrix}$ and  $B=\begin{bmatrix} 1 & 3 & -6 \end{bmatrix}$, Find the value of $B'A'$.

where $A'$ represents transpose of $A$

5). Check whether the matirx $A=\begin{bmatrix} 0 & 2 & 3 \\ 2 & 0 & -4 \\ -3 & 4 & 0 \end{bmatrix}$  is skew symmetric or not.

6). Show that $A-A'$ is also a skew-symmetric matrix.

1). $\left[\begin{array}{rr} 0& 0 \\0 & 0\end{array}\right]$

2) $X=\begin{bmatrix} \dfrac{2}{5}& -\dfrac{12}{5} \\-\dfrac{11}{5} & 3\end{bmatrix}$ and $\begin{bmatrix} \dfrac{2}{5} & \dfrac{13}{5} \\ \dfrac{14}{15} & -2 \end{bmatrix}$

3). $k=1$

4). $\begin{bmatrix} -2 & 4 & 5 \\ -6 & 12 & 15 \\ 12 & -24 &-30 \end{bmatrix}$

5). Yes it is skew symmetric.