Add, subtract, multiply matrices, and compute determinants, transposes, and inverses for up to 3x3 matrices.
Quick Answer:Enter values into the matrix grids, select an operation, and see the result matrix instantly. Supports 1x1 up to 3x3 matrices.
Operation Result
Calculating...
Result Matrix
Determinant of A
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Matrix Dimensions
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Matrix operations are the backbone of modern computing. Graphics transformations (rotation, scaling, translation) in games and 3D software are all matrix multiplications. Neural networks in AI are essentially chains of matrix multiplications with activation functions. Understanding matrices gives you insight into how computers process images, train AI models, and solve systems of equations simultaneously.
Matrix multiplication is done by taking the dot product of rows from the first matrix with columns of the second matrix. For an m x n matrix multiplied by an n x p matrix, the result is an m x p matrix. Each element (i,j) of the result is the sum of products of elements from row i of matrix A and column j of matrix B. Note that matrix multiplication is not commutative: A*B does not necessarily equal B*A.
The determinant is a scalar value computed from a square matrix. For a 2x2 matrix [[a,b],[c,d]], det = ad - bc. The determinant is zero when the matrix is singular, meaning it has no inverse, its rows or columns are linearly dependent, and the system of equations it represents has no unique solution. A non-zero determinant means the matrix is invertible.
The inverse of matrix A, denoted A^(-1), is the matrix such that A * A^(-1) = I (identity matrix). It exists only for square matrices with a non-zero determinant. For a 2x2 matrix, the inverse is (1/det) * [[d,-b],[-c,a]]. For larger matrices, methods like Gauss-Jordan elimination or the adjugate method are used. Singular matrices (det = 0) have no inverse.