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Select the matrix size, input the values, and the determinant calculator will calculate the determinant with detailed steps.
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The determinant calculator simplifies finding determinants for square matrices up to 5×5 in size. You can select the size of the matrix and input either real or complex numbers to evaluate the determinant along with step-by-step calculations.
A determinant is a scalar value obtained from the elements of a square matrix. It has important properties in linear algebra and geometry, such as measuring how much a linear transformation defined by the matrix stretches or compresses space. A positive determinant preserves orientation, while a negative determinant reverses it. Determinants are denoted as det(A), |A|, or det A.
Determinants can be calculated in several ways, including row reduction, cofactor expansion, and diagonal multiplication (for small matrices). This calculator allows you to compute determinants for 2×2, 3×3, 4×4, and 5×5 matrices without manual effort.
For matrix A = [[a, b], [c, d]]:
\( \text{det} A = ad - bc \)
Example:
\( A = \begin{bmatrix} 4 & 12 \\ 2 & 7 \end{bmatrix} \)
\( \text{det} A = (4)(7) - (12)(2) = 28 - 24 = 4 \)
For matrix A = [[a, b, c], [d, e, f], [g, h, i]], using cofactor expansion along the first column:
\( \text{det} A = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - d \begin{vmatrix} b & c \\ h & i \end{vmatrix} + g \begin{vmatrix} b & c \\ e & f \end{vmatrix} \)
Example:
\( A = \begin{bmatrix} 2 & 0 & 3 \\ 1 & 4 & 1 \\ 0 & 4 & 7 \end{bmatrix} \)
\( \text{det} A = 2\begin{vmatrix}4 & 1 \\ 4 & 7\end{vmatrix} - 1\begin{vmatrix}0 & 3 \\ 4 & 7\end{vmatrix} + 0\begin{vmatrix}0 & 3 \\ 4 & 1\end{vmatrix} \)
\( = 2(28 - 4) - 1(0 - 12) + 0 = 48 + 12 = 36 \)
For matrix A = [[a, b, c, d], [e, f, g, h], [i, j, k, l], [m, n, o, p]], expand along the first column:
\( \text{det} A = a \begin{vmatrix} f & g & h \\ j & k & l \\ n & o & p \end{vmatrix} - e \begin{vmatrix} b & c & d \\ j & k & l \\ n & o & p \end{vmatrix} + i \begin{vmatrix} b & c & d \\ f & g & h \\ n & o & p \end{vmatrix} - m \begin{vmatrix} b & c & d \\ f & g & h \\ j & k & l \end{vmatrix} \)
Then calculate the 3×3 determinants using the formula above.
Example:
\( A = \begin{bmatrix} 1 & 8 & 7 & 2 \\ 2 & 4 & 3 & 8 \\ 1 & 4 & 3 & 2 \\ 1 & 4 & 9 & 6 \end{bmatrix} \)
Step-by-step cofactor expansion along the first column gives: \( \text{det} A = -80 \)
For matrix A = [[a, b, c, d, e], [f, g, h, i, j], [k, l, m, n, o], [p, q, r, s, t], [u, v, w, x, y]], expand along the first column:
\( \text{det} A = a \begin{vmatrix} g & h & i & j \\ l & m & n & o \\ q & r & s & t \\ v & w & x & y \end{vmatrix} - f \begin{vmatrix} b & c & d & e \\ l & m & n & o \\ q & r & s & t \\ v & w & x & y \end{vmatrix} + k \begin{vmatrix} b & c & d & e \\ g & h & i & j \\ q & r & s & t \\ v & w & x & y \end{vmatrix} - p \begin{vmatrix} b & c & d & e \\ g & h & i & j \\ l & m & n & o \\ q & r & s & t \end{vmatrix} + u \begin{vmatrix} b & c & d & e \\ g & h & i & j \\ l & m & n & o \\ q & r & s & t \end{vmatrix} \)
Use the 4×4 determinant formula recursively to find the result.
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