Matrix Calculator

To multiply two matrices A and B, the number of columns in A must equal the number of rows in B. Each element in the result is calculated by taking the dot product of a row from A and a column from B. For element (i,j), multiply each element in row i of A by the corresponding element in column j of B, then sum all products. The resulting matrix has dimensions (rows of A) x (columns of B).

A determinant is a scalar value that can only be calculated for square matrices (same number of rows and columns). For a 2x2 matrix [[a,b],[c,d]], the determinant is ad-bc. For larger matrices, we use cofactor expansion or row reduction. A non-zero determinant means the matrix is invertible, while a zero determinant means the matrix is singular.

A matrix is invertible (has an inverse) if and only if it is square and its determinant is non-zero. The inverse of matrix A, denoted A^(-1), satisfies A * A^(-1) = I (identity matrix). Matrices with zero determinant are called singular and have no inverse. The inverse is useful for solving systems of linear equations.