Determinant Calculator
Calculate the determinant of a 2x2 or 3x3 matrix instantly, with the full cofactor-expansion working shown.
Determinant Calculator
Calculator
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Formula: 2x2: det = ad - bc | 3x3: det = a(ei-fh) - b(di-fg) + c(dh-eg)
Additional inputs: h (row 3, col 2, for 3x3), i (row 3, col 3, for 3x3).
Worked example — det = -2
Formula
2x2: det = ad - bc | 3x3: det = a(ei-fh) - b(di-fg) + c(dh-eg)
The determinant is a scalar value computed from a square matrix. For 2x2: ad-bc. For 3x3: cofactor expansion along the first row. Determinant = 0 means the matrix is singular (no inverse, zero volume).
Worked Examples
Example 1: 2x2 matrix
Problem:|1 2; 3 4|
Solution:det = (1)(4) - (2)(3) = 4 - 6 = -2
Result:det = -2
Example 2: 3x3 matrix with a zero determinant (singular)
Problem:|1 2 3; 4 5 6; 7 8 9|
Solution:det = 1(5×9-6×8) - 2(4×9-6×7) + 3(4×8-5×7) = 1(45-48) - 2(36-42) + 3(32-35) = -3 + 12 - 9 = 0
Result:det = 0 (singular matrix, no inverse — rows are linearly dependent)
Frequently Asked Questions
What does the determinant represent geometrically?
For a 2x2 matrix, |det| is the area of the parallelogram formed by the column vectors. For 3x3, |det| is the volume of the parallelepiped. Negative det means orientation is reversed.
When is the determinant zero?
When rows/columns are linearly dependent (one row is a multiple of another), or the system of equations has no unique solution. The matrix is then called singular.
How is the determinant used in solving systems of linear equations?
The determinant is central to Cramer's Rule, a method for solving systems of n linear equations with n unknowns. For a system Ax = b, each variable xi is found by replacing the i-th column of matrix A with the vector b and dividing the resulting determinant by the determinant of A. This requires the determinant of A to be nonzero. While elegant, Cramer's Rule is computationally expensive for large systems, and practical solvers like Gaussian elimination or LU decomposition are preferred for numerical computation in software.
What are the key properties of determinants?
Important determinant properties include: det(AB) = det(A) times det(B) for square matrices of the same size; det(A-transpose) = det(A); det(A-inverse) = 1/det(A) for invertible matrices; multiplying a row by scalar k multiplies the determinant by k; swapping two rows changes the sign of the determinant; adding a multiple of one row to another does not change the determinant; and the determinant of a triangular matrix equals the product of its diagonal entries. These properties underlie practical determinant computation methods like cofactor expansion and row reduction.
How is the 3x3 determinant formula derived from cofactor expansion?
Cofactor expansion along the first row breaks a 3x3 determinant into three 2x2 determinants: det = a·(ei−fh) − b·(di−fg) + c·(dh−eg), where each 2x2 'minor' is formed by deleting the row and column of the corresponding entry (a, b, or c). The alternating +/−/+ signs come from the checkerboard pattern of cofactor signs, (−1)^(row+col). This same technique extends to compute determinants of any size, though it becomes computationally expensive for large matrices compared to row-reduction methods.
What is the connection between the determinant and the matrix inverse?
A square matrix has an inverse if and only if its determinant is nonzero, and the inverse is computed as A⁻¹ = adj(A) / det(A), where adj(A) is the adjugate (transpose of the cofactor matrix). This is why checking the determinant first is the standard first step before attempting to invert a matrix — a zero determinant means no inverse exists at all.
How large can matrices get before the determinant is impractical to compute by cofactor expansion?
Cofactor expansion has factorial time complexity — an n×n matrix requires roughly n! multiplications, so a 10×10 matrix already needs over 3.6 million operations by this method. In practice, computer algebra systems and numerical libraries use LU decomposition (Gaussian elimination with partial pivoting) to compute determinants in cubic time, O(n³), which is dramatically faster for anything larger than a 3x3 or 4x4 matrix.
What is the difference between the determinant and the trace of a matrix?
The trace is the sum of the diagonal entries, while the determinant is the alternating cofactor sum. For a 2x2 matrix both appear in the characteristic polynomial, lambda^2 - (trace)lambda + det = 0, whose roots are the eigenvalues. In general the determinant equals the product of all eigenvalues and the trace equals their sum, so the determinant detects a zero eigenvalue (singularity) while the trace does not.
How does row reduction let you find a determinant without cofactor expansion?
Gaussian elimination reduces a matrix to upper triangular form, and a triangular matrix's determinant is simply the product of its diagonal pivots. You track the effect of each operation: adding a multiple of one row to another leaves the determinant unchanged, swapping two rows flips its sign, and scaling a row by k multiplies it by k. Undoing those bookkeeping factors on the pivot product recovers the original determinant.
Why is the determinant only defined for square matrices?
The determinant measures how a linear map scales volume in the space it acts on, which only makes sense when the domain and codomain have the same dimension — that is, when the matrix is n x n. A 2x3 matrix maps 3D vectors into a plane, so there is no consistent volume scale factor to report. For rectangular matrices the analogous quantities are singular values or the Gram determinant det(A-transpose A).
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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