Determinant Calculator

About the Determinant Calculator

The determinant condenses a square matrix into a single number that answers several important questions at once. Is the matrix invertible? Does the linear system it represents have a unique solution? By what factor does the transformation stretch areas or volumes, and does it flip orientation? All of these are read from the determinant.

This calculator computes determinants of square matrices of any practical size. It chooses the method a textbook would use: the cross-multiplication rule for $2 \times 2$ matrices, expansion along the first row for $3 \times 3$ matrices, and elimination to upper-triangular form for $4 \times 4$ and larger. Fractions and decimals are converted to exact fractions, so the answer is exact.

What is a determinant?

The determinant is a number $\det(A)$ , also written $|A|$ , defined for every square matrix. For a $2 \times 2$ matrix it is $ad - bc$ . For larger matrices it is defined recursively by cofactor expansion, or equivalently as a signed sum over all permutations of the columns.

Geometrically, $|\det(A)|$ is the factor by which the linear map $A$ scales area in two dimensions or volume in three. A negative determinant means the map reverses orientation, like a reflection, and a zero determinant means the map squashes space into a lower dimension.

How to use the Determinant Calculator

Enter a square matrix with spaces between entries and semicolons between rows, for example 2 -3 1; 2 0 -1; 1 4 5.
Press Find determinant.
Read the method used for your matrix size and each intermediate value.
Check whether the determinant is zero to decide if the matrix is invertible.

Formulas

2×2 determinant

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Cofactor expansion (first row)

\det(A) = \sum_{j} (-1)^{1+j}\, a_{1j}\, M_{1j}

Product rule

\det(AB) = \det(A)\,\det(B)

Triangular matrix

\det(U) = u_{11}\,u_{22}\cdots u_{nn}

Worked example: 3×3 matrix

The example finds the determinant of the $3 \times 3$ matrix with rows $(2, -3, 1)$ , $(2, 0, -1)$ and $(1, 4, 5)$ by expanding along the first row. Each first-row entry is multiplied by the $2 \times 2$ minor left after deleting its row and column, with the signs $+$ , $-$ , $+$ .

The minors evaluate to 4, 11 and 8, so the determinant is $2 \cdot 4 - (-3) \cdot 11 + 1 \cdot 8 = 8 + 33 + 8 = 49$ . A nonzero value means this matrix is invertible.

The solution below is generated by the same engine that powers the calculator above.

Final answer

\det A = 49

How we read your problem

Find the determinant of a 3×3 matrix.

\det \begin{bmatrix}2 & -3 & 1 \\ 2 & 0 & -1 \\ 1 & 4 & 5\end{bmatrix}

Step-by-step solution

Read the 3×3 matrix
$A = \begin{bmatrix}2 & -3 & 1 \\ 2 & 0 & -1 \\ 1 & 4 & 5\end{bmatrix}$
Expand along the first row (cofactor expansion)
$\det A = (2)\begin{vmatrix}0 & -1 \\ 4 & 5\end{vmatrix} - (-3)\begin{vmatrix}2 & -1 \\ 1 & 5\end{vmatrix} + (1)\begin{vmatrix}2 & 0 \\ 1 & 4\end{vmatrix}$
Evaluate each 2×2 minor
$M_{11} = 4,\quad M_{12} = 11,\quad M_{13} = 8$
Combine with the alternating signs + − +
$\det A = (2)(4) - (-3)(11) + (1)(8) = 49$

Formulas used

2×2 determinant
$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$
Cofactor expansion
$\det A = \sum_{j} (-1)^{1+j} a_{1j} M_{1j}$

The determinant is 49, which is non-zero, so the matrix is invertible and its rows are linearly independent.

Verification

Computed exactly with fractions; row reduction gives the same value 49.

Detailed explanation

Cofactor expansion

The minor $M_{ij}$ is the determinant of the matrix left after deleting row $i$ and column $j$ . The cofactor attaches the sign $(-1)^{i+j}$ , which follows a checkerboard pattern starting with $+$ in the top-left corner. Expanding along any row or column gives the same result, so by hand it pays to choose a row or column containing zeros.

Cofactor expansion is ideal for $3 \times 3$ matrices but grows very quickly: a $4 \times 4$ expansion needs four $3 \times 3$ determinants, and a $10 \times 10$ expansion would need millions of terms. That is why the calculator switches method for larger matrices.

Row reduction to triangular form

Adding a multiple of one row to another does not change the determinant, swapping two rows changes its sign, and scaling a row scales the determinant by the same factor. Using these rules, the calculator eliminates the entries below each pivot until the matrix is upper triangular, then multiplies the diagonal entries. For the $4 \times 4$ matrix with rows $(1, 0, 2, 0)$ , $(0, 3, 0, 1)$ , $(2, 0, 1, 0)$ and $(0, 1, 0, 2)$ this gives $-15$ .

What a zero determinant means

A determinant of zero means the rows are linearly dependent: at least one row is a combination of the others. The matrix has no inverse, and the system $A\mathbf{x} = \mathbf{b}$ has either no solution or infinitely many. The matrix with rows $(1, 2, 3)$ , $(4, 5, 6)$ and $(7, 8, 9)$ is the classic example; its determinant is 0 because the middle row is the average of the other two.

Determinants also appear in Cramer's rule for solving small systems, in the characteristic polynomial $\det(A - \lambda I) = 0$ whose roots are the eigenvalues, and in the change-of-variables formula for multiple integrals.

Common mistakes

Computing $ad + bc$ or $ab - cd$ instead of $ad - bc$ .
Forgetting the alternating signs in cofactor expansion.
Using the wrong minor by deleting the wrong row or column.
Ignoring the sign change after swapping two rows during row reduction.
Trying to take the determinant of a non-square matrix.

Tips

Expand along the row or column with the most zeros.
Factor a common number out of a row first; the determinant scales by that same number.
If two rows are equal or proportional, the determinant is 0 without further work.
Use $\det(A^{T}) = \det(A)$ to expand along a column when it is easier.
Use the inverse matrix calculator when the determinant is nonzero and you need $A^{-1}$ .
Use the eigenvalue calculator, which builds on $\det(A - \lambda I)$ .

Frequently asked questions

How do you find the determinant of a 2×2 matrix?

Multiply the main diagonal entries and subtract the product of the other diagonal: for rows (a, b) and (c, d) the determinant is ad − bc.

How do you find the determinant of a 3×3 matrix?

Expand along a row: multiply each entry by its 2×2 minor, apply the signs +, −, + and add. The calculator shows each minor and the final sum.

What method is used for 4×4 and larger matrices?

Row reduction to upper-triangular form, followed by multiplying the diagonal entries. This is much faster than cofactor expansion for large matrices.

What does a determinant of zero mean?

The matrix is singular: its rows are linearly dependent, it has no inverse and the corresponding linear system does not have a unique solution.

Can a determinant be negative?

Yes. A negative determinant means the transformation reverses orientation. Its absolute value is still the area or volume scaling factor.

Can I find the determinant of a non-square matrix?

No. Determinants are only defined for square matrices, and the calculator reports the size of a non-square input.

Are fractions supported?

Yes. Fractions and decimals are converted to exact fractions, so the determinant is exact.

Find your determinant

Enter a square matrix above to see the determinant and every step of the method used.

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