Question

Evaluate :

$\begin{gathered} \sf A = \begin{bmatrix} 2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7 \end{bmatrix} \end{gathered}$
Also find minor and cofactor elements in the 2nd row of determinant and verify.

$\small\sf{(a) \: -a_{21} .M_{21} + a_{22}.M_{22} - a_{23} .M_{23} = value \: of \: A }$
$\small\sf{(b) \: a_{21}.C_{21}+ a_{22}.C_{22} + a_{23}.C_{23}= Value of A}$
$\small\rm{Where \: M_{21},M_{22},M_{23} \: are \: minor \: of \: a_{21},a_{22},a_{23} }$
$\small\rm{And \: C_{21},C_{22},C_{23} \: are \: Cofactor \: of \: a_{21},a_{22},a_{23}. }$
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Answer 1

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

Given matrix is

$\rm :\longmapsto\:A = \: \begin{gathered} \sf \begin{bmatrix} 2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7 \end{bmatrix} \end{gathered}$

Now, The value of matrix A is called | A |

So,

$\rm :\longmapsto\: |A|$

$\rm \:  =  \:2\begin{array}{|cc|}\sf 0 &\sf 4 \\ \sf 5 &\sf - 7 \\\end{array} + 3\begin{array}{|cc|}\sf 6 &\sf 4 \\ \sf 1 &\sf - 7 \\\end{array} + 5\begin{array}{|cc|}\sf 6 &\sf 0 \\ \sf 1 &\sf 5 \\\end{array}$

$\rm \:  =  \:2(0 - 20) + 3( - 42 - 4) + 5(30 - 0)$

$\rm \:  =  - 40 - 138 + 150$

$\rm \:  =  \: - \: 28$

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

Now, To verify

$\sf{(a) \: -a_{21} .M_{21} + a_{22}.M_{22} - a_{23} .M_{23} = value \: of \: A }$

Now,

$\red{\rm :\longmapsto\:a_{21} = 6}$

$\red{\rm :\longmapsto\:a_{22} = 0}$

$\red{\rm :\longmapsto\:a_{23} = 4}$

$\red{\rm :\longmapsto\:M_{21} = \begin{array}{|cc|}\sf - 3 &\sf 5 \\ \sf 5 &\sf - 7 \\\end{array} = 21 - 25 = - 4}$

$\red{\rm :\longmapsto\:M_{22} = \begin{array}{|cc|}\sf 2 &\sf 5 \\ \sf 1 &\sf - 7 \\\end{array} = - 14 - 5 = - 19}$

$\red{\rm :\longmapsto\:M_{23} = \begin{array}{|cc|}\sf 2 &\sf - 3 \\ \sf 1 &\sf 5 \\\end{array} = 10 + 3= 13}$

So,

$\rm :\longmapsto\: -a_{21} .M_{21} + a_{22}.M_{22} - a_{23} .M_{23}$

$\rm \:  =  \: - 6( - 4) + 0( - 19) - 4(13)$

$\rm \:  =  \: 24 + 0 - 52$

$\rm \:  =  \: - 28$

$\rm \:  =  \: |A|$

Hence, Verified

Now, To verify

$\sf{(b) \: a_{21}.C_{21}+ a_{22}.C_{22} + a_{23}.C_{23}= Value of A}$

We know,

$\boxed{ \: \: \: \: \bf \:C_{ij} = {( - 1)}^{i + j}M_{ij} \: \: \: \: }$

Now,

$\red{\rm :\longmapsto\:C_{21} = {( - 1)}^{2 + 1} \begin{array}{|cc|}\sf - 3 &\sf 5 \\ \sf 5 &\sf - 7 \\\end{array} = - (21 - 25) = 4}$

$\red{\rm :\longmapsto\:C_{22} = {( - 1)}^{2 + 2} \begin{array}{|cc|}\sf 2 &\sf 5 \\ \sf 1 &\sf - 7 \\\end{array} = - 14 - 5 = - 19}$

$\red{\rm :\longmapsto\:C_{23} = {( - 1)}^{2 + 3} \begin{array}{|cc|}\sf 2 &\sf - 3 \\ \sf 1 &\sf 5 \\\end{array} = - (10 + 3)= - 13}$

So,

$\rm :\longmapsto\: \: a_{21}.C_{21}+ a_{22}.C_{22} + a_{23}.C_{23}$

$\rm \:  =  \:6(4) - 0(19) + 4( - 13)$

$\rm \:  =  \:24 - 0 - 52$

$\rm \:  =  \: - 28$

$\rm \:  =  \: |A|$

Hence, Verified