# Test: Matrices II - Ambitious

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Question 1:   Find $x$ and $y$: $\left[\begin{array}{cc}-2& 3\\ 4& -5\end{array}\right]\left[\begin{array}{cc}x& 2\\ 7& y\end{array}\right]=\left[\begin{array}{cc}23& 5\\ -13& -7\end{array}\right]$
$x=-10.5,y=-1\frac{3}{4}$
$x=-2,y=-5$
$x=23,y=7$
$x=-1,y=3$
Question 2:   Find $p$ and $q$: $\left[\begin{array}{cc}3& 5\\ 4& -1\end{array}\right]\left[\begin{array}{cc}p& 2\\ 2& q\end{array}\right]=\left[\begin{array}{cc}7& 31\\ -6& 3\end{array}\right]$
$p=-1,q=5$
$p=3,q=-1$
$p=4,q=2$
$p=7,q=3$
Question 3:   Find $\mathrm{det}A$ if $A=\left[\begin{array}{cc}5& 7\\ -6& -4\end{array}\right]$
$|A|=62$
$|A|=17$
$|A|=22$
$|A|=-20$
Question 4:   Which of the matrices is singular?
$D=\left[\begin{array}{cc}3& 1\\ 1& 0\end{array}\right]$
$B=\left[\begin{array}{cc}-7& 2\\ 7& 2\end{array}\right]$
$C=\left[\begin{array}{cc}-5& 0\\ 0& 5\end{array}\right]$
$A=\left[\begin{array}{cc}-3& 6\\ -4& 8\end{array}\right]$
Question 5:   Which of the following matrices is the transpose of $B$, if

$B=\left[\begin{array}{cccc}2& 3& -1& 1\\ 5& 0& 8& 3\\ -6& 5& 2& 5\\ 8& 7& 0& -1\end{array}\right]$?
${B}^{T}=\left[\begin{array}{cccc}1& -1& 3& 2\\ 3& 8& 0& 5\\ 5& 2& 5& -6\\ -1& 0& 7& 8\end{array}\right]$
${B}^{T}=\left[\begin{array}{cccc}2& 5& -6& 8\\ 3& 0& 5& 7\\ -1& 8& 2& 0\\ 1& 3& 5& -1\end{array}\right]$
${B}^{T}=\left[\begin{array}{cccc}8& 7& 0& -1\\ -6& 5& 2& 5\\ 5& 0& 8& 3\\ 2& 3& -1& 1\end{array}\right]$
${B}^{T}=\left[\begin{array}{cccc}1& 3& 5& -1\\ -1& 8& 2& 0\\ 3& 0& 5& 7\\ 2& 5& -6& 8\end{array}\right]$
Question 6:   Find the transpose ${A}^{T}$, if $A=\left[\begin{array}{ccc}3& 0& 6\\ 1& -1& 2\\ 5& 2& 7\end{array}\right]$
${A}^{T}=\left[\begin{array}{ccc}3& 1& 5\\ 0& -1& 2\\ 6& 2& 7\end{array}\right]$
${A}^{T}=\left[\begin{array}{ccc}7& 2& 6\\ 2& -1& 0\\ 5& 1& 3\end{array}\right]$
${A}^{T}=\left[\begin{array}{ccc}5& 2& 7\\ 1& -1& 2\\ 3& 0& 6\end{array}\right]$
${A}^{T}=\left[\begin{array}{ccc}6& 0& 3\\ 2& -1& 1\\ 7& 2& 5\end{array}\right]$
Question 7:   Which of the following is inverse of $A$, if $A=\left[\begin{array}{cc}3& 5\\ -2& -1\end{array}\right]$?
${A}^{-1}=\left[\begin{array}{cc}\frac{3}{7}& \frac{2}{7}\\ -\frac{5}{7}& -\frac{1}{7}\end{array}\right]$
${A}^{-1}=\left[\begin{array}{cc}-1& -5\\ 2& 3\end{array}\right]$
${A}^{-1}=\left[\begin{array}{cc}-3& -2\\ 1& 5\end{array}\right]$
${A}^{-1}=\left[\begin{array}{cc}-\frac{1}{7}& -\frac{5}{7}\\ \frac{2}{7}& \frac{3}{7}\end{array}\right]$
Question 8:   Find the inverse of $A$, if $A=\left[\begin{array}{cc}-1& 4\\ 2& -3\end{array}\right]$
${A}^{-1}=\left[\begin{array}{cc}\frac{3}{5}& \frac{4}{5}\\ \frac{2}{5}& \frac{1}{5}\end{array}\right]$
${A}^{-1}=\left[\begin{array}{cc}-\frac{2}{5}& -\frac{1}{5}\\ -\frac{3}{5}& -\frac{4}{5}\end{array}\right]$
${A}^{-1}=\left[\begin{array}{cc}-2& -1\\ -3& -4\end{array}\right]$
${A}^{-1}=\left[\begin{array}{cc}-3& -4\\ -2& -1\end{array}\right]$
Question 9:   Solve: $8{A}^{-1}$, if $A=\left[\begin{array}{cc}-2& 2\\ -1& -3\end{array}\right]$
$\left[\begin{array}{cc}1& -2\\ -3& -2\end{array}\right]$
$\left[\begin{array}{cc}-\frac{3}{64}& -\frac{1}{24}\\ \frac{1}{64}& -\frac{1}{24}\end{array}\right]$
$\left[\begin{array}{cc}-\frac{3}{8}& -\frac{1}{4}\\ \frac{1}{8}& -\frac{1}{4}\end{array}\right]$
$\left[\begin{array}{cc}-3& -2\\ 1& -2\end{array}\right]$
Question 10:   Find $2{C}^{-1}$, if $C=\left[\begin{array}{cc}2& -2\\ 5& -5\end{array}\right]$
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
Not defined
$\left[\begin{array}{cc}-5& 2\\ 2& 5\end{array}\right]$