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.