Matrix determinant calculator
Compute the determinant of a 2x2 or 3x3 matrix.
What this calculator covers
Use this matrix determinant calculator to reduce a square 2x2 or 3x3 matrix to its determinant.
The walkthrough keeps the active matrix and determinant rule visible so the sign pattern and multiplication order are easier to audit.
Frequently asked questions
- What does a determinant of zero mean?
- A determinant of zero means the matrix is singular — its rows or columns are linearly dependent. In practical terms, a singular matrix has no inverse, and a system of linear equations represented by that matrix has either no solution or infinitely many solutions.
- What is the sign pattern used in 3x3 cofactor expansion?
- When expanding along the first row, the cofactors alternate in sign as plus, minus, plus for the three entries. This alternating pattern comes from the definition of the determinant and must be applied correctly for the result to be accurate.
- Why does the calculator only support 2x2 and 3x3 matrices?
- These two sizes cover the most common homework and textbook scenarios and allow the full calculation steps to be shown clearly. Larger matrices require recursive or row-reduction methods that produce much longer intermediate outputs.
- Can the determinant be negative?
- Yes — a negative determinant is completely valid. It indicates that the transformation described by the matrix includes a reflection, reversing the orientation of the coordinate system. The magnitude still gives information about scaling.
Tool
Run the calculation
Result
RESULT · DETERMINANT
â„–218
Primary result
14
The determinant of the 2x2 matrix [4, 6] [3, 8] is 14.
- Matrix
- [4, 6] [3, 8]
- Expansion
- (4 x 8) - (6 x 3)
- Determinant
- 14
Step-by-step solution
- 1.Write the active matrix entries as [4, 6] [3, 8].
- 2.Use ad - bc for the 2x2 determinant.
- 3.Evaluate (4 x 8) - (6 x 3) to get 14.
Walkthrough
Visual walkthrough
Determinants condense a square matrix into one signed scalar that reflects area, volume, and invertibility behavior.
01
Read the active square matrix
[4, 6] [3, 8]
Only the selected 2x2 or 3x3 block participates in the determinant calculation.
02
Apply the determinant rule
(4 x 8) - (6 x 3)
For a 2x2 matrix, multiply the main diagonal and subtract the off diagonal product.
03
Read the scalar result
That single number is the determinant of the selected matrix.
14