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Question:
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Match the following elements of $\left[\begin{array}{ccc}1 & -1 & 0 \\ 0 & 4 & 2 \\ 3 & -4 & 6\end{array}\right]$ with their co-factors and choose the correct answer.
$\begin{array}{lcll} & \text { Element } & & \text { Co-factor } \\ \text { A. } & -1 & \text { (1) } & -2 \\ \text { B. } & 1 & (2) & 32 \\ \text { C. } & 3 & (3) & 4 \\ \text { D. } & 6 & \text { (4) } & 6 \\ & & \text { (5) } & -6\end{array}$
Options:
$\begin{array}{lcll} & \text { Element } & & \text { Co-factor } \\ \text { A. } & -1 & \text { (1) } & -2 \\ \text { B. } & 1 & (2) & 32 \\ \text { C. } & 3 & (3) & 4 \\ \text { D. } & 6 & \text { (4) } & 6 \\ & & \text { (5) } & -6\end{array}$
Solution:
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Verified Answer
The correct answer is:
$\begin{array}{llll}\text { A } & \text { B } & \text { C } & \text { D } \\ 4 & 2 & 1 & 3\end{array}$
Let $A=\left[\begin{array}{ccc}1 & -1 & 0 \\ 0 & 4 & 2 \\ 3 & -4 & 6\end{array}\right]$
Cofactor of $(-1)=-\left|\begin{array}{ll}0 & 2 \\ 3 & 6\end{array}\right|=6$
Cofactor of $(1)=\left|\begin{array}{cc}4 & 2 \\ -4 & 6\end{array}\right|=24+8=32$
Cofactor of $(3)=\left|\begin{array}{cc}-1 & 0 \\ 4 & 2\end{array}\right|=-2-0=-2$
Cofactor of $(6)=\left|\begin{array}{cc}1 & -1 \\ 0 & 4\end{array}\right|=4-0=4$
$\therefore$ Option (3) is correct.
Cofactor of $(-1)=-\left|\begin{array}{ll}0 & 2 \\ 3 & 6\end{array}\right|=6$
Cofactor of $(1)=\left|\begin{array}{cc}4 & 2 \\ -4 & 6\end{array}\right|=24+8=32$
Cofactor of $(3)=\left|\begin{array}{cc}-1 & 0 \\ 4 & 2\end{array}\right|=-2-0=-2$
Cofactor of $(6)=\left|\begin{array}{cc}1 & -1 \\ 0 & 4\end{array}\right|=4-0=4$
$\therefore$ Option (3) is correct.
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