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If \(A=\left[\begin{array}{ll}1 & 3 \\ 3 & 2 \\ 2 & 5\end{array}\right]\) and \(B=\left[\begin{array}{cc}-1 & -2 \\ 0 & 5 \\ 3 & 1\end{array}\right]\) and \(\mathrm{A}+\mathrm{B}-\mathrm{D}=0\) (zero matrix), then \(\mathrm{D}\) matrix will be -
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\(\left[\begin{array}{ll}0 & 1 \\ 3 & 7 \\ 5 & 6\end{array}\right]\)
Let \(D=\left[\begin{array}{ll}a & b \\ c & d \\ e & f\end{array}\right]\)
\(\begin{aligned} & \therefore A+B-C=\left[\begin{array}{ll}1 & 3 \\ 3 & 2 \\ 2 & 5\end{array}\right]+\left[\begin{array}{cc}-1 & -2 \\ 0 & 5 \\ 3 & 1\end{array}\right]-\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{d} \\ \mathrm{e} & \mathrm{f}\end{array}\right] \\ & \Rightarrow\left[\begin{array}{ll}1-1-\mathrm{a} & 3-2-\mathrm{b} \\ 3+0-\mathrm{c} & 2+5-\mathrm{d} \\ 2+3-\mathrm{e} & 5+1-\mathrm{f}\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0 \\ 0 & 0\end{array}\right]\end{aligned}\)
\(\begin{array}{ll}-\mathrm{a}=0 \Rightarrow \mathrm{a}=0,1-\mathrm{b}=0 \Rightarrow \mathrm{b}=1, \\ 3-\mathrm{c}=0 \Rightarrow \mathrm{c}=3, & 7-\mathrm{d}=0 \Rightarrow \mathrm{d}=7, \\ 5-\mathrm{e}=0 \Rightarrow \mathrm{e}=5, & 6-\mathrm{f}=0 \Rightarrow \mathrm{f}=6,\end{array}\)
\(\therefore D=\left[\begin{array}{ll}0 & 1 \\ 3 & 7 \\ 5 & 6\end{array}\right]\)
\(\begin{aligned} & \therefore A+B-C=\left[\begin{array}{ll}1 & 3 \\ 3 & 2 \\ 2 & 5\end{array}\right]+\left[\begin{array}{cc}-1 & -2 \\ 0 & 5 \\ 3 & 1\end{array}\right]-\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{d} \\ \mathrm{e} & \mathrm{f}\end{array}\right] \\ & \Rightarrow\left[\begin{array}{ll}1-1-\mathrm{a} & 3-2-\mathrm{b} \\ 3+0-\mathrm{c} & 2+5-\mathrm{d} \\ 2+3-\mathrm{e} & 5+1-\mathrm{f}\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0 \\ 0 & 0\end{array}\right]\end{aligned}\)
\(\begin{array}{ll}-\mathrm{a}=0 \Rightarrow \mathrm{a}=0,1-\mathrm{b}=0 \Rightarrow \mathrm{b}=1, \\ 3-\mathrm{c}=0 \Rightarrow \mathrm{c}=3, & 7-\mathrm{d}=0 \Rightarrow \mathrm{d}=7, \\ 5-\mathrm{e}=0 \Rightarrow \mathrm{e}=5, & 6-\mathrm{f}=0 \Rightarrow \mathrm{f}=6,\end{array}\)
\(\therefore D=\left[\begin{array}{ll}0 & 1 \\ 3 & 7 \\ 5 & 6\end{array}\right]\)
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