Enter the elements of your 2×2, 3×3, or 4×4 matrix and the Matrix Determinant Calculator computes the determinant value for you. Select your matrix size, fill in each cell, and get the scalar result — useful for checking invertibility, solving linear systems, and more.
Determinant
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Matrix Size
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Invertible?
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Trace (Sum of Diagonal)
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A determinant is a scalar value computed from a square matrix that encodes important properties of the matrix. It is used to determine whether a matrix is invertible, solve systems of linear equations, and find eigenvalues. A matrix is invertible if and only if its determinant is non-zero.
For a 2×2 matrix [[a, b], [c, d]], the determinant is simply ad − bc. Multiply the main diagonal elements and subtract the product of the off-diagonal elements.
The 3×3 determinant is computed using cofactor expansion (Laplace expansion) along any row or column. For the first row, det = a(ei−fh) − b(di−fg) + c(dh−eg), where the letters represent the matrix elements in order. This tool performs that calculation for you automatically.
A 4×4 determinant is computed by expanding along a row or column into four 3×3 cofactor determinants, each multiplied by the corresponding element and sign (+/−). The process involves recursive cofactor expansion, which this calculator handles automatically.
A determinant of zero means the matrix is singular — it is not invertible. The rows (or columns) are linearly dependent, meaning the matrix does not have a unique inverse and the corresponding system of linear equations either has no solution or infinitely many solutions.
Yes, the determinant can be any real number including negative values. A negative determinant indicates that the matrix transformation reverses orientation (e.g. a reflection), while a positive determinant preserves orientation. The absolute value of the determinant represents the scaling factor of the transformation.
The trace is the sum of the main diagonal elements of a square matrix. It equals the sum of the matrix's eigenvalues and is a useful companion value to the determinant when analyzing matrix properties. Both the determinant and trace appear in the characteristic polynomial used to find eigenvalues.
Yes. Swapping any two rows or two columns changes the sign of the determinant. Multiplying a row by a scalar k multiplies the determinant by k. These properties are fundamental to row reduction techniques used to compute determinants of large matrices.