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 $\det(A) = 5$.

Number domains

The number domain selector changes what an entry is. The same operations run over each, because the engine is written against an abstract field:

• Fractions — exact rationals (the default).
• Complex numbers — exact Gaussian rationals $a + b\,i$ with rational $a, b$. Closed under $+,-,\times,\div$ (divide by the conjugate), so the determinant, inverse, RREF and solver stay exact. Enter cells like 3+2i, -i, or 1/2 - 3/4 i.
• Symbolic — entries may contain variables (a, 2x, x^2, (x+1)/(x-1)). Values are multivariate rational functions over $\mathbb{Q}$, which form a field, so $\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$ and $\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}^{-1} = \begin{bmatrix} 1/a & 0 \\ 0 & 1/b \end{bmatrix}$ come out symbolically. Pivots are taken in generic position (a symbolically-nonzero entry is treated as nonzero), and results are simplified conservatively — never wrongly.

Decompositions

• LU — $PA = LU$ with a lower-triangular $L$, an upper-triangular $U$, and a permutation $P$ recording row swaps. Exact over any domain.
• QR — $A = QR$ with orthonormal $Q$ and upper-triangular $R$ (Gram–Schmidt). Needs square roots, so it is numeric (approximate).
• Cholesky — for a symmetric positive-definite $A$, $A = LL^{\mathsf T}$. Also numeric, and reported as not-SPD when no such factor exists.

Spectrum

The characteristic polynomial $\det(\lambda I - A)$ is computed exactly by the Faddeev–LeVerrier recurrence over any domain. Eigenvalues are its roots: exact when they are rational, otherwise found numerically (so a rotation matrix $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ reports $\pm i$, flagged approximate). Each eigenvector spans the null space of $A - \lambda I$, exact for rational $\lambda$ and numeric otherwise.

Results that cannot be exact are marked approximate so the distinction is never hidden.