MCQ
If $\triangle=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}$ and $A_{ij}$ is cofactors of $a_{ij},$ then value of $\triangle$ is given by:
  • A
    $a_{11}A_{31} + a_{12}A_{32} + a_{13}A_{33}$
  • B
    $a_{11}A_{11} + a_{12}A_{21} + a_{13}A_{31}$
  • C
    $a_{21}A_{11} + a_{22}A_{12} + a_{23}A_{13}$
  • $a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31}$

Answer

Correct option: D.
$a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31}$
We know that:
$\triangle =$ Sum of the product of the elements of a column $($or a row$)$ with their corresponding cofactors
$\therefore\triangle = \text{a}_{11}\text{A}_{11} +\text{a}_{21}\text{A}_{21} + \text{a}_{31}\text{A}_{31}$
Hence, the value of $\triangle$ is given by the expression given in alternative $d$. the correct answer is $d$.

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