Matrix Determinant Calculator
The matrix determinant calculator finds the determinant of a 2x2 or 3x3 square matrix and tells you whether the matrix is invertible or singular. The determinant is a single scalar value computed from all the entries of a square matrix. It carries deep geometric meaning: the absolute value of the determinant of a 2x2 matrix equals the area of the parallelogram formed by its row vectors, and for a 3x3 matrix it equals the volume of the parallelepiped. In algebra, the determinant tells you whether a matrix has an inverse. If the determinant is non-zero, the matrix is invertible (non-singular) and the associated system of linear equations has a unique solution. If the determinant is zero, the matrix is singular, its rows are linearly dependent, and no inverse exists. For a 2x2 matrix with entries a, b in the first row and c, d in the second row, the determinant is simply ad minus bc. For a 3x3 matrix, the calculator uses cofactor expansion along the first row, breaking the determinant into three 2x2 sub-determinants with alternating signs. You choose the matrix size, enter all the entries, and the calculator returns the determinant, the invertibility verdict, and the full cofactor expansion steps so you can follow the working. The tool is ideal for students in linear algebra courses, engineers solving systems of equations, and programmers implementing matrix operations. The results are mathematically exact for integer inputs.
How it works
For a 2x2 matrix [[a,b],[c,d]], the determinant is ad - bc. This is the simplest case and can be remembered as the product of the main diagonal minus the product of the anti-diagonal. For a 3x3 matrix [[a,b,c],[d,e,f],[g,h,i]], cofactor expansion along the first row gives: det = a(ei - fh) - b(di - fg) + c(dh - eg). Each element in the first row multiplies the determinant of the 2x2 submatrix formed by deleting its row and column, with signs following the checkerboard pattern (+, -, +). The matrix is invertible when the determinant is not zero, and singular (no inverse) when it is zero.
Worked example
For the default 2x2 matrix with entries [4, 7] and [2, 6]: det = (4 x 6) - (7 x 2) = 24 - 14 = 10. Since 10 is not zero, the matrix is invertible. For the default 3x3 matrix with rows [1,2,3], [4,5,6], [7,8,9]: det = 1(5x9 - 6x8) - 2(4x9 - 6x7) + 3(4x8 - 5x7) = 1(45-48) - 2(36-42) + 3(32-35) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0. The matrix is singular because the rows are linearly dependent.
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