Matrix Calculator

What this tool does

The matrix calculator works over the rational numbers: every entry is kept as an exact fraction, so results are exact — 1/3 never drifts to 0.3333. Enter a matrix, pick an operation, and (where it helps) read the worked steps.

Core operations

For matrices $A$ and $B$ of compatible sizes:

Matrix multiplication is not commutative — $AB$ and $BA$ usually differ — so both are offered.

Gaussian elimination

Row reduction underlies rank, the determinant, the inverse, and solving systems. Each step is one of three elementary row operations: swap two rows, scale a row, or add a multiple of one row to another. Reducing to reduced row-echelon form (a leading 1 in each pivot column, zeros elsewhere in that column) exposes the rank and the solution set.

Determinant

Reducing $A$ to upper-triangular form by row operations, the determinant is the product of the diagonal, with a sign flip for each row swap:

A zero on the diagonal after elimination means $\det(A) = 0$ and $A$ is singular.

Inverse

Augment $A$ with the identity and run Gauss–Jordan elimination: $[\,A \mid I\,] \to [\,I \mid A^{-1}\,]$. If the left block cannot reach $I$, the matrix is singular and has no inverse.

Solving $Ax = b$

Row-reduce the augmented matrix $[\,A \mid b\,]$. A pivot in the $b$ column means no solution. Otherwise the pivots fix the basic variables; any free columns give a null-space basis, so the full solution is one particular vector plus any combination of those basis vectors.

Worked example

With $A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}$ and $b = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, elimination gives the unique solution $x = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$, and $\det(A) = 5$.