Matrix Equality and Compatibility Checker

Math Linear Algebra • Matrices and Systems of Equations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Compare two matrices \(A\) and \(B\): check equality (with optional tolerance) and check compatibility for addition/subtraction and multiplication (both \(AB\) and \(BA\)).

Rows of A (m)

Cols of A (n)

Rows of B (p)

Cols of B (q)

Display decimals

Equality tolerance

Steps

Play scans cell-by-cell (debug view) and stops automatically. Drag to pan • wheel to zoom • double-click to reset.

Matrix \(A\)

2×2

Matrix \(B\)

2×2

Ready

Equality

—

Add/Sub

—

Multiply AB

—

Multiply BA

—

Max |A−B|

—

Mismatches

—

Debug scan (A vs B)

Highlights the scanned cell \((i,j)\) in both matrices. Mismatch cells are marked.

A B current cell mismatch

Enter matrices \(A\) and \(B\) and click “Calculate”.

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Theory

Matrix equality

Two matrices \(A\) and \(B\) are equal if and only if they have the same dimensions and every corresponding entry matches: \[ A=B \quad \Longleftrightarrow \quad A\in\mathbb{R}^{m\times n},\, B\in\mathbb{R}^{m\times n} \text{ and } a_{ij}=b_{ij}\ \forall i,j. \]

In real software, matrix entries are often floating-point numbers, so “exact equality” can fail due to rounding. A common approach is to use a small tolerance \(\text{tol}\) and declare matrices equal when \[ \max_{i,j}\,|a_{ij}-b_{ij}| \le \text{tol}. \] This calculator supports both exact equality (\(\text{tol}=0\)) and tolerance-based equality.

Compatibility for addition and subtraction

Matrix addition and subtraction are defined only when the matrices have the same size: \[ A\pm B \text{ is defined } \Longleftrightarrow A\in\mathbb{R}^{m\times n},\ B\in\mathbb{R}^{m\times n}. \] If the sizes differ, you cannot add or subtract entry-by-entry because the grids do not line up.

Compatibility for multiplication

Matrix multiplication has a different size rule. If \(A\in\mathbb{R}^{m\times n}\) and \(B\in\mathbb{R}^{p\times q}\), then the product \(AB\) is defined exactly when the inner dimensions match: \[ AB \text{ is defined } \Longleftrightarrow n=p. \] When it is defined, the result has size \(m\times q\).

Note that \(AB\) and \(BA\) are different questions. It is common that \(AB\) exists but \(BA\) does not, or vice versa. This calculator checks both directions and reports the resulting dimensions when compatible.

Debug matrix tip

Dimension errors are one of the most frequent causes of mistakes in linear algebra work and in code. Before attempting a computation, it helps to confirm the operation is valid: check sizes for \(A\pm B\), check inner dimensions for \(AB\), then (if needed) check equality entry-by-entry.

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