# 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

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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 $\left|\begin{array}{cc}a & b\\c & d\end{array}\right| = a d - b c$.

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

The determinant of the matrix equals $2$A.