Find the cofactor of b 3 in the following matrix Δ: \(= \left[ {\

$Find the cofactor of b 3 in the following matrix Δ: \(= \left[ {\$

| Find the cofactor of b₃in the following matrix Δ: $= \left[ {\begin{array}{*{20}{c}} a_1&{ b_1 }&{c_1}\\ a_2&b_2&{ c_2}\\ a_3&b_3&c_3 \end{array}} \right]$

A. $\left| {\begin{array}{*{20}{c}} a_1 &c_1\\ a_2&c_2 \end{array}} \right| $

B. - b3

C. $-\left| {\begin{array}{*{20}{c}} a_1 &c_1\\ a_2&c_2 \end{array}} \right| $

D. - 1

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Right Answer is: C

SOLUTION

Cofactor of $a_1 = {A_{11}} = \left| {\begin{array}{*{20}{c}} b_2 &c_2\\ b_3&c_3 \end{array}} \right| $

Cofactor of $b_1 = {A_{12}} = -\left| {\begin{array}{*{20}{c}} a_2 &c_2\\ a_3&c_3 \end{array}} \right| $

Cofactor of $c_1 = {A_{13}} = \left| {\begin{array}{*{20}{c}} a_2 &b_2\\ a_3&b_3 \end{array}} \right| $

Cofactor of $a_2 = {A_{21}} = -\left| {\begin{array}{*{20}{c}} b_1 &c_1\\ b_3&c_3 \end{array}} \right| $

Cofactor of $b_2 = {A_{22}} = \left| {\begin{array}{*{20}{c}} a_1 &c_1\\ a_3&c_3 \end{array}} \right| $

Cofactor of $c_2 = {A_{23}} = -\left| {\begin{array}{*{20}{c}} a_1 &b_1\\ a_3&b_3 \end{array}} \right| $

Cofactor of $a_3 = {A_{31}} = \left| {\begin{array}{*{20}{c}} b_1 &c_1\\ b_2&c_2 \end{array}} \right| $

Cofactor of $b_3 = {A_{32}} =- \left| {\begin{array}{*{20}{c}} a_1 &c_1\\ a_2&c_2 \end{array}} \right| $

Cofactor of $c_3 = {A_{33}} = \left| {\begin{array}{*{20}{c}} a_1 &b_1\\ a_2&b_2 \end{array}} \right| $