Matrix Calculator
Solve matrix operations from 1x1 up to 4x4. Compute matrix multiplication, transposes, determinants, and inverses with step-by-step expansions.
Computed Matrix Output
multiply CompletedLinear Algebra Workings Log
Understanding Matrix Operations
A matrix is a rectangular array of numbers arranged in rows and columns. In linear algebra, matrices are used to represent systems of linear equations, track states in quantum mechanics, rotate coordinates in 3D gaming engines, and scale weights in neural networks.
Matrix Multiplication (A × B)
Multiplying two matrices is not a simple cell-by-cell multiplication. Instead, the elements of rows in Matrix A are multiplied by corresponding columns in Matrix B and summed up. This means the number of columns in A must strictly equal the number of rows in B.
Matrix Inversion (A-1)
The inverse of a square matrix A is another matrix A-1 such that A × A-1 = I, where I is the identity matrix. If a matrix has a determinant of zero, it is called singular, meaning it lacks any multiplicative inverse.
Frequently Asked Questions About Matrices
When can two matrices be multiplied?
Two matrices can be multiplied if and only if the number of columns in the first matrix (Matrix A) is equal to the number of rows in the second matrix (Matrix B). The resulting matrix will have the number of rows of Matrix A and the number of columns of Matrix B.
What is a singular matrix?
A singular matrix is a square matrix that has a determinant of zero. Singular matrices do not have a multiplicative inverse, meaning they cannot be inverted.
How is a matrix determinant calculated?
For a 2x2 matrix, the determinant is ad - bc. For higher dimensions (3x3 and 4x4), it is calculated recursively using cofactor expansion, where elements of a row or column are multiplied by the determinants of their corresponding submatrices, alternating signs.
Can this handle 4x4 matrix operations?
Yes. Solve addition, multiplication, determinants, transposes, and inverses up to 4x4 dimensions.