Question 47 of 472MediumMCQ
Let A=[aij] be a 3×3 matrix such that A[[∣010]]∣=[[∣001]]∣,A[[∣413]]∣=[[∣010]]∣ and A[[∣212]]∣=[[∣100]]∣, then a23 equals :
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Official solution
\begin{gathered} A=[[|\begin{array}{|c|}{a}_{11} \quad {a}_{12} \quad {a}_{13} \\ {a}_{21} \quad {a}_{22} \quad {a}_{23} \\ {a}_{31} \quad {a}_{32} \quad {a}_{33}\end{array}]]| \\ A[[|\begin{array}{|c|}0 \\ 1 \\ 0\end{array}]]|=[[|\begin{array}{|c|}0 \\ 0 \\ 1\end{array}]]|\Rightarrow [[|\begin{array}{|c|}{a}_{12} \\ {a}_{22} \\ {a}_{32}\end{array}]]|=[[|\begin{array}{|c|}0 \\ 1 \\ 0\end{array}]]|\Rightarrow \begin{gathered} {a}_{22}=0;{a}_{12}=0 \\ {a}_{32}=1 \end{gathered}
\begin{gathered} A[[|\begin{array}{|c|}4 \\ 1 \\ 3\end{array}]]|=[[|\begin{array}{|c|}0 \\ 1 \\ 0\end{array}]]|\Rightarrow 4{a}_{11}+{a}_{12}+3{a}_{13}=0 \\ 4{a}_{21}+{a}_{22}+3{a}_{23}=1\Rightarrow 4{a}_{21}+3{a}_{23}=1 \\ 4{a}_{31}+{a}_{32}+3{a}_{33}=0 \\ A[[|\begin{array}{|c|}2 \\ 1 \\ 2\end{array}]]|=[[|\begin{array}{|c|}1 \\ 0 \\ 0\end{array}]]|\Rightarrow 2{a}_{11}+{a}_{12}+2{a}_{13}=1 \\ 2{a}_{21}+{a}_{22}+2{a}_{23}=0\Rightarrow {a}_{21}+{a}_{23}=0 \\ 2{a}_{31}+{a}_{32}+2{a}_{33}=0 \end{gathered} \\ -4{a}_{23}+3{a}_{23}=1\Rightarrow {a}_{23}=-1 \end{gathered}
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