If the cofactors of the elements 3,7 and 6 of the matrix $\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & -1 & 7 \\…

If the cofactors of the elements 3,7 and 6 of the matrix $\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6\end{array}\right]$ are a, b and c respectively, then $\left[\begin{array}{lll}\mathrm{a} & \mathrm{b} & \mathrm{c}\end{array}\right]\left[\begin{array}{l}1 \\ 4 \\ 2\end{array}\right]+\left[\begin{array}{lll}\mathrm{a} & \mathrm{b} & \mathrm{c}\end{array}\right]\left[\begin{array}{l}3 \\ 7 \\ 6\end{array}\right]=$
  1. $-1$
  2. $1$
  3. $0$
  4. $3$

Solution

In the given matrix $\begin{aligned} & C_{13}=(-1)^{1+3}(16+2) \Rightarrow a=18 \\ & C_{23}=(-1)^{2+3}(4-4) \Rightarrow b=0 \\ & C_{33}=(-1)^{3+3}(-1-8) \Rightarrow c=-9 \end{aligned}$ Now, $\begin{aligned} & {\left[\begin{array}{lll} 18 & 0 & -9 \end{array}\right]\left[\begin{array}{l} 1 \\ 4 \\ 2 \end{array}\right]=\left[\begin{array}{lll} 18 & 0 & 9 \end{array}\right]\left[\begin{array}{l} 3 \\ 7 \\ 6 \end{array}\right]} \\ & =\left[\begin{array}{lll} 18 & 0 & -9 \end{array}\right]\left[\begin{array}{c} 4 \\ 11 \\ 8 \end{array}\right]=[72-72]=[0]=0 \end{aligned}$

Asked in: AP EAMCET 2023 (15 May Shift 2)

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