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Determinant of $$$\left[\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-3 & 2 & 1\end{array}\right]$$$

The calculator will find the determinant of the square $$$3$$$x$$$3$$$ matrix $$$\left[\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-3 & 2 & 1\end{array}\right]$$$, with steps shown.

Related calculator: Cofactor Matrix Calculator

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Your Input

Calculate $$$\left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-3 & 2 & 1\end{array}\right|$$$.

Solution

Subtract row $$$1$$$ from row $$$3$$$: $$$R_{3} = R_{3} - R_{1}$$$.

$$$\left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-3 & 2 & 1\end{array}\right| = \left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-1 & -1 & 0\end{array}\right|$$$

Expand along column $$$3$$$:

$$$\left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-1 & -1 & 0\end{array}\right| = \left(1\right) \left(-1\right)^{1 + 3} \left|\begin{array}{cc}7 & -4\\-1 & -1\end{array}\right| + \left(0\right) \left(-1\right)^{2 + 3} \left|\begin{array}{cc}-2 & 3\\-1 & -1\end{array}\right| + \left(0\right) \left(-1\right)^{3 + 3} \left|\begin{array}{cc}-2 & 3\\7 & -4\end{array}\right| = \left|\begin{array}{cc}7 & -4\\-1 & -1\end{array}\right|$$$

The determinant of a 2x2 matrix is $$$\left|\begin{array}{cc}a & b\\c & d\end{array}\right| = a d - b c$$$.

$$$\left|\begin{array}{cc}7 & -4\\-1 & -1\end{array}\right| = \left(7\right)\cdot \left(-1\right) - \left(-4\right)\cdot \left(-1\right) = -11$$$

Answer

$$$\left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-3 & 2 & 1\end{array}\right| = -11$$$A