Loading calculator...
Loading calculator...
Perform matrix operations on 2x2 and 3x3 matrices: calculate determinant, find inverse, and multiply matrices.
Matrix A
Matrix Formulas:
2x2 Determinant: |A| = ad - bc
2x2 Inverse: (1/|A|) x [[d,-b],[-c,a]]
3x3 Determinant: cofactor expansion along first row
Matrix Multiply: C[i][j] = sum(A[i][k] * B[k][j])
The determinant is a scalar value computed from a square matrix. It indicates whether the matrix is invertible (non-zero determinant = invertible). For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
A matrix has no inverse (it is singular) when its determinant equals zero. This means the matrix transforms space in a way that collapses dimensions.
Matrix multiplication C = A x B combines transformations. The element C[i][j] is the dot product of row i of A and column j of B. Note: A x B is generally not equal to B x A.
The identity matrix I has 1s on the main diagonal and 0s elsewhere. It is the multiplicative identity: A x I = I x A = A. The inverse of A satisfies A x A^(-1) = I.
Matrices are fundamental in linear algebra, used for solving systems of equations, computer graphics transformations, machine learning (neural networks), physics simulations, and many engineering applications.