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1. ⇒ (MHT CET 2021 20th September Evening Shift )

The co-factors of the elements of second column of $[\begin{array}{ccc} 1 & - 1 & 2 \\ 3 & 2 & 1 \\ - 1 & 3 & 4 \end{array}]$ are

A. $- 13, 6, 5$

B. $13, 5, 6$

C. $13, - 6, - 5$

D. $- 13, - 6, 5$

Correct answer option is (A)

Required cofactors are as follows :

$\begin{aligned} A_{12} = (- 1)^{1 + 2} | \begin{array}{cc} 3 & 1 \\ - 1 & 4 \end{array} | = - (12 + 1) = - 13 \\ A_{22} = (- 1)^{2 + 2} | \begin{array}{cc} 1 & 2 \\ - 1 & 4 \end{array} | = (4 + 2) = 6 \\ A_{32} = (- 1)^{3 + 2} | \begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} | = - (1 - 6) = 5 \end{aligned}$

2. ⇒ (MHT CET 2021 20th September Morning Shift )

If $A = [\begin{matrix} 3 & 2 & 4 \\ 1 & 2 & 1 \\ 3 & 2 & 6 \end{matrix}]$ and A $_{i j}$ are cofactors of the elements a $_{i j}$ of A, then $a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13}$ is equal to

A. 8

B. 6

C. 4

D. 0

Correct answer option is (A)

$a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} = | A |$

$∴$ $| A | = | \begin{matrix} 3 & 2 & 4 \\ 1 & 2 & 1 \\ 3 & 2 & 6 \end{matrix} |$

$= 3 (12 - 2) - 2 (6 - 3) + 4 (2 - 6)$

$= 30 - 6 - 16 = 8$

⇒Matrices