Question

Find the matrix A such that
(i) $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]A=\left[\begin{array}{ccc}3& 3& 5\\ 1& 0& 1\end{array}\right]$

(ii) $A\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]=\left[\begin{array}{ccc}-7& -8& -9\\ 2& 4& 6\end{array}\right]$

(iii) $\left[\begin{array}{c}4\\ 1\\ 3\end{array}\right]A=\left[\begin{array}{ccc}-4& 8& 4\\ -1& 2& 1\\ -3& 6& 3\end{array}\right]$

(iv) $\left[\begin{array}{ccc}2& 1& 3\end{array}\right]\left[\begin{array}{ccc}-1& 0& -1\\ -1& 1& 0\\ 0& 1& 1\end{array}\right]\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]=A$

Answer 1

​$$\begin{gathered} \left(\mathrm{i}\right) \\ \mathrm{Let}A=\left[\begin{array}{ccc}x& y& z\\ a& b& c\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{ccc}x& y& z\\ a& b& c\end{array}\right]=\left[\begin{array}{ccc}3& 3& 5\\ 1& 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}x+a& y+b& z+c\\ 0+a& 0+b& 0+c\end{array}\right]=\left[\begin{array}{ccc}3& 3& 5\\ 1& 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}x+a& y+b& z+c\\ a& b& c\end{array}\right]=\left[\begin{array}{ccc}3& 3& 5\\ 1& 0& 1\end{array}\right] \\ \mathrm{The}\mathrm{corresponding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \Rightarrow x+a=3...\left(1\right) \\ y+b=3...\left(2\right) \\ z+c=5...\left(3\right) \\ \Rightarrow a=1,b=0\mathrm{and}c=1 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}a\mathrm{in}\mathrm{eq}.\left(1\right),\mathrm{we}\mathrm{get} \\ x+1=3 \\ \Rightarrow x=3-1 \\ \therefore x=2 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}b\mathrm{in}\mathrm{eq}.\left(2\right),\mathrm{we}\mathrm{get} \\ y+b=3 \\ \Rightarrow y+0=3 \\ \therefore y=3 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}c\mathrm{in}\mathrm{eq}.\left(3\right),\mathrm{we}\mathrm{get} \\ z+1=5 \\ \Rightarrow z=5-1 \\ \therefore z=4 \\ \therefore A=\left[\begin{array}{ccc}2& 3& 4\\ 1& 0& 1\end{array}\right] \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{Let}A=\left[\begin{array}{cc}w& x\\ y& z\end{array}\right] \\ \Rightarrow A\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]=\left[\begin{array}{ccc}-7& -8& -9\\ 2& 4& 6\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}w& x\\ y& z\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]=\left[\begin{array}{ccc}-7& -8& -9\\ 2& 4& 6\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}w+4x& 2w+5x& 3w+6x\\ y+4z& 2y+5z& 3y+6z\end{array}\right]=\left[\begin{array}{ccc}-7& -8& -9\\ 2& 4& 6\end{array}\right] \\ \mathrm{The}\mathrm{correspnding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \therefore 3w+6x=-9...\left(1\right) \\ y+4z=2 \\ y=2-4z...\left(2\right) \\ w+4x=-7 \\ \Rightarrow w=-7-4x...\left(3\right) \\ 2y+5z=4...\left(4\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}w\mathrm{in}\mathrm{eq}.\left(1\right),\mathrm{we}\mathrm{get} \\ 3\left(-7-4x\right)+6x=-9 \\ \Rightarrow -21-12x+6x=-9 \\ \Rightarrow -6x=12 \\ \Rightarrow x=-2 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}x\mathrm{in}\mathrm{eq}.\left(3\right),\mathrm{we}\mathrm{get} \\ w=-7-4\left(-2\right) \\ \Rightarrow w=-7+8 \\ \Rightarrow w=1 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}y\mathrm{in}\mathrm{eq}.\left(4\right),\mathrm{we}\mathrm{get} \\ 2\left(2-4z\right)+5z=4 \\ \Rightarrow 4-8z+5z=4 \\ \Rightarrow 4-3z=4 \\ \Rightarrow -3z=0 \\ \Rightarrow z=0 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}z\mathrm{in}\mathrm{eq}.\left(2\right),\mathrm{we}\mathrm{get} \\ y=2-4\left(0\right) \\ \Rightarrow y=2 \\ \therefore A=\left[\begin{array}{cc}1& -2\\ 2& 0\end{array}\right] \end{gathered}$$


$$\begin{gathered} \left(\mathrm{iii}\right) \\ \mathrm{Let}A=\left[\begin{array}{ccc}x& y& z\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}4\\ 1\\ 3\end{array}\right]\left[\begin{array}{ccc}x& y& z\end{array}\right]=\left[\begin{array}{ccc}-4& 8& 4\\ -1& 2& 1\\ -3& 6& 3\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}4x& 4y& 4z\\ x& y& z\\ 3x& 3y& 3z\end{array}\right]=\left[\begin{array}{ccc}-4& 8& 4\\ -1& 2& 1\\ -3& 6& 3\end{array}\right] \\ \mathrm{The}\mathrm{corresponding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \Rightarrow 4x=-4...\left(1\right) \\ 4y=8...\left(2\right) \\ 4z=4...\left(3\right) \\ \Rightarrow x=-1,y=2\mathrm{and}z=1 \\ \therefore A=\left[\begin{array}{ccc}-1& 2& 1\end{array}\right] \end{gathered}$$


$$\begin{gathered} \left(\mathrm{iv}\right)\mathrm{Let}A=\left[\begin{array}{c}x\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}2& 1& 3\end{array}\right]\left[\begin{array}{ccc}-1& 0& -1\\ -1& 1& 0\\ 0& 1& 1\end{array}\right]\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]=A \\ \Rightarrow \left[\begin{array}{ccc}2& 1& 3\end{array}\right]\left[\begin{array}{ccc}-1& 0& -1\\ -1& 1& 0\\ 0& 1& 1\end{array}\right]\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]=\left[\begin{array}{c}x\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}-2-1+0& 0+1+3& -2+0+3\end{array}\right]\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]=\left[\begin{array}{c}x\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}-3& 4& 1\end{array}\right]\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]=\left[\begin{array}{c}x\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}-3+0-1\end{array}\right]=\left[\begin{array}{c}x\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}-4\end{array}\right]=\left[\begin{array}{c}x\end{array}\right] \\ \mathrm{The}\mathrm{corresponding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \therefore x=-4 \\ \therefore A=\left[-4\right] \end{gathered}$$