Overview
A matrix determinant is a number that captures its geometric and algebraic properties. For a 3×3 matrix it can be computed by expanding a row or column with cofactors. A zero determinant means the matrix is singular and not invertible.
Formula
|A| = a(ei − fh) − b(di − fg) + c(dh − eg).
How it works
Enter the nine entries
Arrange them in three rows and three columns.
Apply the expansion formula
Use the first row and its cofactors.
Compute the determinant
Add the three terms with their signs.
Check the result
If the determinant is zero, the matrix is not invertible.
Worked examples
Example 1
The matrix [[1, 2, 3], [0, 1, 4], [5, 6, 0]] has determinant 1.
Example 2
A matrix with two identical rows has determinant zero.
Frequently asked questions
Does row order affect the determinant?
Swapping two rows changes the sign but not the absolute value.
What does the determinant mean geometrically?
It represents the volume of the parallelepiped formed by the row vectors.
Can I expand along any row?
Yes, expanding along any row or column gives the same result.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.