Matrix Determinant Calculator

The calculator will find the determinant of the matrix (2x2, 3x3, 4x4 etc.) using the cofactor expansion, with steps shown.

Related calculator: Cofactor Matrix Calculator

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Your Input

Calculate $$$\left|\begin{array}{ccc}1 & 2 & 2\\0 & 5 & 7\\1 & 1 & 1\end{array}\right|$$$.

Solution

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

$$$\left|\begin{array}{ccc}1 & 2 & 2\\0 & 5 & 7\\1 & 1 & 1\end{array}\right| = \left|\begin{array}{ccc}1 & 2 & 2\\0 & 5 & 7\\0 & -1 & -1\end{array}\right|$$$

Expand along column $$$1$$$:

$$$\left|\begin{array}{ccc}1 & 2 & 2\\0 & 5 & 7\\0 & -1 & -1\end{array}\right| = \left(1\right) \left(-1\right)^{1 + 1} \left|\begin{array}{cc}5 & 7\\-1 & -1\end{array}\right| + \left(0\right) \left(-1\right)^{2 + 1} \left|\begin{array}{cc}2 & 2\\-1 & -1\end{array}\right| + \left(0\right) \left(-1\right)^{3 + 1} \left|\begin{array}{cc}2 & 2\\5 & 7\end{array}\right| = \left|\begin{array}{cc}5 & 7\\-1 & -1\end{array}\right|$$$

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

$$$\left|\begin{array}{cc}5 & 7\\-1 & -1\end{array}\right| = \left(5\right)\cdot \left(-1\right) - \left(7\right)\cdot \left(-1\right) = 2$$$

Answer

The determinant of the matrix equals $$$2$$$A.