Evaluate | ccc 2 & -3 & 5 6 & 0 & 4 1 & 5 & -7 | and cofactors of elements in the 2^ nd determinant and verify: i. -a_ 21 M_ 21 +a_ 22 M_ 22 -a_ 23 M_ 23 = value of A a_ 21 C_ 21 +a_ 22 C_ 22 +a_ 23 C_ 23 - value of A where M_ 21 , M_ 22 , M_ 23 are minors of a_ 21 , a_ 22 , a_ 23 and C_ 21 , C_ 22 , C_ 23 are cofactors of a_ 21 , a_ 22 , a_ 23 .

& A= | ccc 2 & -3 & 5 6 & 0 & 4 1 & 5 & -7 | =2 | cc 0 & 4 5 & -7 |-(-3) | cc 6 & 4 1 & -7 |+5 | cc 6 & 0 1 & 5 | =2(0-20)+3(-42-4)+5(30-0)=2(-20)+3(-46)+5(30) =2(0-20)+3(-42-4)+5(30-0)=2(-20)+3(-46)+5(30) =-40-138+150 =-28 Here, | lll a_ 11 & a_ 12 & a_ 13 a_ 21 & a_ 22 & a_ 23 a_ 31 & a_ 32 & a_ 33 |= | ccc 2 & -3 & 5 6 & 0 & 4 1 & 5 & -7 | & M_ 21 = | cc -3 & 5 5 & -7 |=21-25=-4 C_ 21 =(-1)^ 2+1 M_ 21 =(-1)(-4)=4 M_ 22 = | cc 2 & 5 1 & -7 |=-14-5=-19 C_ 22 =(-1)^ 2+2 M_ 22 =(1)(-19)=-19 M_ 23 = | cc 2 & -3 1 & 5 |=10+3=13 C_ 23 =(-1)^ 2+3 M_ 23 =(-1)(13)=-13 -a_ 21 M_ 21 +a_ 22 M_ 22 -a_ 23 M_ 23 =-(6)(-4)+(0)(-19)-(4)(13) =24+0-52 =-28 -a_ 21 M_ 21 +a_ 22 M_ 22 -a_ 23 M_ 23 = value of A ii. a_ 21 C_ 21 +a_ 22 C_ 22 +a_ 23 C_ 23 =(6)(4)+(0)(-19)+(4)(-13) =24+0-52 . =-28 a_ 21 C_ 21 +a_ 22 C_ 22 +a_ 23 C_ 23 = value of A

Evaluate | ccc 2 & -3 & 5 6 & 0 & 4 1 & 5 & -7 | and cofactors of…

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Question

Evaluate $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ and cofactors of elements in the $2^{nd}$ determinant and verify:
$i. -a_{21} \cdot M_{21}+a_{22} \cdot M_{22}-a_{23} \cdot M_{23}=$ value of $A a_{21} \cdot C_{21}+a_{22} \cdot C_{22}+a_{23} \cdot C_{23}-$ value of $A$ where $M_{21}, M_{22}, M_{23}$ are minors of $a_{21}, a_{22}, a_{23}$ and $C_{21}, C_{22}, C_{23}$ are cofactors of $a_{21}, a_{22}, a_{23}$.

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Answer

$\begin{aligned} & A=\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right| \end{aligned} $
$ =2\left|\begin{array}{cc}0 & 4 \\ 5 & -7\end{array}\right|-(-3)\left|\begin{array}{cc}6 & 4 \\ 1 & -7\end{array}\right|+5\left|\begin{array}{cc}6 & 0 \\ 1 & 5\end{array}\right|$
$ =2(0-20)+3(-42-4)+5(30-0)=2(-20)+3(-46)+5(30)$
$ =2(0-20)+3(-42-4)+5(30-0)=2(-20)+3(-46)+5(30)$
$ =-40-138+150$
$=-28$
Here, $\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|=\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$
$\begin{aligned} & \mathrm{M}_{21}=\left|\begin{array}{cc}-3 & 5 \\ 5 & -7\end{array}\right|=21-25=-4 \end{aligned} $
$ \mathrm{C}_{21}=(-1)^{2+1} \mathrm{M}_{21}=(-1)(-4)=4 $
$ \mathrm{M}_{22}=\left|\begin{array}{cc}2 & 5 \\ 1 & -7\end{array}\right|=-14-5=-19 $
$ \mathrm{C}_{22}=(-1)^{2+2} \mathrm{M}_{22}=(1)(-19)=-19 $
$ \mathrm{M}_{23}=\left|\begin{array}{cc}2 & -3 \\ 1 & 5\end{array}\right|=10+3=13$
$ \mathrm{C}_{23}=(-1)^{2+3} \mathrm{M}_{23}=(-1)(13)=-13$
$ -\mathrm{a}_{21} \cdot M_{21}+\mathrm{a}_{22} \cdot M_{22}-\mathrm{a}_{23} \cdot M_{23}$
$ =-(6)(-4)+(0)(-19)-(4)(13)$
$ =24+0-52$
$ =-28$
$ -a_{21} \cdot M_{21}+a_{22} \cdot M_{22}-a_{23} \cdot M_{23}=$ value of $A$
$ \text { ii. } a_{21} \cdot C_{21}+a_{22} \cdot C_{22}+a_{23} \cdot C_{23}$
$ =(6)(4)+(0)(-19)+(4)(-13)$
$ =24+0-52 .$
$ =-28$
$ a_{21} \cdot C_{21}+a_{22} \cdot C_{22}+a_{23} \cdot C_{23}=$ value of $A$

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