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For a 2×2 matrix, multiply the main diagonal and subtract the other diagonal: \(\det\begin{bmatrix}a & b\\ c & d\end{bmatrix} = ad – bc\). For a 3×3 matrix, use cofactor expansion along one row. For 4×4 and larger, row reduce to a triangular matrix and multiply the diagonal. If the determinant is 0, the matrix has no inverse.
How to Enter a Matrix
Type one row at a time. Put a space or a comma between numbers, and a semicolon between rows. Fractions and decimals both work.
| Matrix | Type this |
|---|---|
| \(\begin{bmatrix}3 & 8\\ 4 & 6\end{bmatrix}\) | 3 8; 4 6 |
| \(\begin{bmatrix}2 & 1 & 3\\ 0 & 4 & 5\\ 1 & 0 & 6\end{bmatrix}\) | 2 1 3; 0 4 5; 1 0 6 |
| \(\begin{bmatrix}\tfrac12 & \tfrac13\\ \tfrac14 & \tfrac15\end{bmatrix}\) | 1/2 1/3; 1/4 1/5 |
The calculator picks the fastest method for the size you enter. For a 3×3 matrix it expands along the row with the most zeros, because every zero removes a whole 2×2 determinant from the work.
The 2×2 Determinant in One Line
Example: \(\begin{vmatrix} 3 & 8 \\ 4 & 6 \end{vmatrix} = 3\cdot 6 – 8\cdot 4 = 18 – 32 = -14\).
A negative determinant is fine. The sign only tells you the matrix flips orientation, like a mirror image.
3×3 Determinant: Cofactor Expansion, Step by Step
Let’s find the determinant of \(A = \begin{bmatrix}2 & 1 & 3\\ 0 & 4 & 5\\ 1 & 0 & 6\end{bmatrix}\) by expanding along the first row.
- Write the sign pattern. The signs go \(+\ -\ +\) across the first row.
- Cross out a row and a column for each entry. What is left is a 2×2 “minor.”
\[\det A = 2\begin{vmatrix}4 & 5\\0 & 6\end{vmatrix} – 1\begin{vmatrix}0 & 5\\1 & 6\end{vmatrix} + 3\begin{vmatrix}0 & 4\\1 & 0\end{vmatrix}\] - Find each 2×2 determinant. \(24\), \(-5\) and \(-4\).
- Combine. \(2(24) – 1(-5) + 3(-4) = 48 + 5 – 12 = 41\).
You can expand along any row or any column and you will always get 41. Choosing a row or column with zeros just means less arithmetic.
Bigger Matrices: Use Row Operations
Expanding a 4×4 matrix by cofactors needs four 3×3 determinants, which means twelve 2×2 determinants. Row reduction is much shorter. Turn the matrix into an upper triangular one, then multiply the numbers on the diagonal. Just keep track of how each row operation changes the determinant:
| Row operation | Effect on the determinant |
|---|---|
| Swap two rows | Multiply by \(-1\) |
| Multiply a row by \(k\) | Multiply by \(k\) |
| Add a multiple of one row to another | No change |
The calculator only uses swaps and “add a multiple” steps, then lists every operation it did, so you can follow along with a pencil.
What the Determinant Tells You
- Invertible or not. A square matrix has an inverse exactly when \(\det A \ne 0\). The matrix calculator can find that inverse for you.
- Area and volume. A 2×2 matrix stretches every shape’s area by \(|\det A|\). A 3×3 matrix does the same for volume.
- Systems of equations. If the coefficient matrix has a nonzero determinant, the system has exactly one solution.
- Eigenvalues. The determinant equals the product of the eigenvalues. Check it with the eigenvalue calculator.
- Cross products and Jacobians. The cross product and the Jacobian determinant are both determinants in disguise.
Common mistake: forgetting the minus sign on the middle term of a 3×3 expansion. The signs alternate \(+\,-\,+\) across row 1, but they start with \(-\) on row 2.
Learn the Math Behind the Determinant Calculator
The determinant of a square matrix measures how it scales area or volume. It is zero exactly when the matrix has no inverse.
Frequently Asked Questions
Can a non-square matrix have a determinant?
No. Only square matrices (2×2, 3×3, and so on) have determinants. For other shapes, the matrix calculator can still find the rank or the reduced row echelon form.
What does a determinant of 0 mean?
The matrix is singular. It has no inverse, its rows (and columns) are linearly dependent, and it squashes space into a lower dimension, so areas or volumes become 0.
Is det(AB) equal to det(A) times det(B)?
Yes. The determinant of a product is the product of the determinants. But \(\det(A + B)\) is usually not \(\det A + \det B\).
What is the rule of Sarrus?
It is a shortcut for 3×3 matrices only: copy the first two columns to the right, add the three down-right diagonal products and subtract the three up-right ones. It gives the same answer as cofactor expansion, but it does not work for 4×4 matrices.
Keep Practicing
- Matrix calculator: multiply, invert, and row reduce.
- Eigenvalue and eigenvector calculator: uses \(\det(A – \lambda I) = 0\).
- Wronskian calculator: a determinant built from functions and their derivatives.