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Determinants and Matrices — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDeterminants and Matrices3 Marks
Question
Find the value of determinant expanding along third column $\left|\begin{array}{ccc}-1 & 1 & 2 \\ -2 & 3 & -4 \\ -3 & 4 & 0\end{array}\right|$
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Answer

Here, $\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|=\left|\begin{array}{ccc}-1 & 1 & 2 \\ -2 & 3 & -4 \\ -3 & 4 & 0\end{array}\right|$
Expantion along the third column
$=a_{13} C_{13}+a_{23} C_{23}+a_{33} C_{33}$
$=2 \times(-1)^{1+3}\left|\begin{array}{ll}-2 & 3 \\ -3 & 4\end{array}\right|-4 \times(-1)^{2+3}\left|\begin{array}{ll}-1 & 1 \\ -3 & 4\end{array}\right|+0 \times(-1)^{3+3}\left|\begin{array}{ll}-1 & 1 \\ -2 & 3\end{array}\right|$
$ =2(-8+9)+4(-4+3)+0$
$ =2-4$
$ =-2$

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