Question

Using cofactors of elements of second row, evaluate |A| where A = |(1,-3,2) (-4,5,6) (3,-5,2)|​

Answer 1

$\large\underline{\sf{Solution-}}$

Given matrix is

$\rm :\longmapsto\:A = \begin{gathered}\sf \left[\begin{array}{ccc}1&-3&2\\ - 4&5&6\\3&-5&2\end{array}\right]\end{gathered}$

We know,

$\rm :\longmapsto\: |A| = a_{21}c_{21} + a_{22}c_{22} + a_{23}c_{23}$

Where,

$\rm :\longmapsto\:a_{21} = - 4$

$\rm :\longmapsto\:a_{22} = 5$

$\rm :\longmapsto\:a_{23} = 6$

Now,

$\rm :\longmapsto\:c_{21} = {( - 1)}^{2 + 1}\begin{array}{|cc|}\sf - 3 &\sf 2 \\ \sf - 5 &\sf 2 \\\end{array} = - ( - 6 + 10) = - 4$

$\rm :\longmapsto\:c_{22} = {( - 1)}^{2 + 2}\begin{array}{|cc|}\sf 1 &\sf 2 \\ \sf 3 &\sf 2 \\\end{array} = 2 - 6 = - 4$

$\rm :\longmapsto\:c_{23} = {( - 1)}^{2 + 3}\begin{array}{|cc|}\sf 1 &\sf - 3 \\ \sf 3 &\sf - 5 \\\end{array} = - ( - 5 + 9) = - 4$

So, on substituting the values in

$\rm :\longmapsto\: |A| = a_{21}c_{21} + a_{22}c_{22} + a_{23}c_{23}$

we get ,

$\rm :\longmapsto\: |A| = - 4( - 4) + 5( - 4) + 6( - 4)$

$\rm :\longmapsto\: |A| = 16 - 20 - 24$

$\bf :\longmapsto\: |A| = - 28$

Additional Information :-

Properties of Co factors :-

$\rm :\longmapsto\: |A| = a_{11}c_{11} + a_{12}c_{12} + a_{13}c_{13}$

$\rm :\longmapsto\: |A| = a_{31}c_{31} + a_{32}c_{32} + a_{33}c_{33}$

$\rm :\longmapsto\: a_{31}c_{11} + a_{32}c_{12} + a_{33}c_{13} = 0$

$\rm :\longmapsto\: a_{31}c_{21} + a_{32}c_{22} + a_{33}c_{23} = 0$