Matrix Multiplication Calculator

Math Linear Algebra • Matrices and Systems of Equations

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

Compute matrix multiplication \(C=A\,B\). If \(A\) is \(r_A\times c_A\) and \(B\) is \(r_B\times c_B\), the product exists only when \(c_A=r_B\). Each entry is a row–column dot product: \(c_{ij}=\sum_{k=1}^{c_A} a_{ik}b_{kj}\).

A rows

A cols

B rows

B cols

Display decimals

Multiply result by scalar \(k\): compute \(k(A B)\) Show all steps (can be long for large matrices)

Matrix \(A\)

2×2

Matrix \(B\)

2×2

Result \(C=A B\)

2×2

Ready

Row–column highlight visual

Drag to pan • wheel to zoom • double-click to reset. Play highlights a row of \(A\) and a column of \(B\) for each \(c_{ij}\), fills cells left-to-right, top-to-bottom, then stops.

A B C highlighted row highlighted column active cell

Enter matrices and click “Calculate”.

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Theory

When matrix multiplication is defined

Matrix multiplication is fundamentally different from matrix addition. Addition is entry-wise and requires the same shape, but multiplication combines rows and columns and requires a compatibility condition. If \(A\) is an \(r_A\times c_A\) matrix and \(B\) is an \(r_B\times c_B\) matrix, then the product \(C=A B\) is defined only when \(c_A=r_B\). In that case, the result \(C\) has size \(r_A\times c_B\). This “inner dimension match” ensures that each row of \(A\) has the same length as each column of \(B\), so a dot product can be formed.

The entry formula is \[ c_{ij}=\sum_{k=1}^{c_A} a_{ik}b_{kj}. \] You can read this as: take the \(i\)-th row of \(A\) and the \(j\)-th column of \(B\), multiply matching components, and add them up. That is exactly a row–column dot product.

Row–column dot product interpretation

Suppose \(A\) and \(B\) are both \(2\times 2\). Then \(c_{11}\) uses row 1 of \(A\) and column 1 of \(B\): \[ c_{11}=a_{11}b_{11}+a_{12}b_{21}. \] The “Play” visualization in this calculator highlights exactly those parts: one row in \(A\) and one column in \(B\), and then fills the corresponding entry in \(C\). Repeating for all \((i,j)\) fills the entire product matrix.

This is also the reason multiplication is not commutative in general. Even when both \(AB\) and \(BA\) exist, the row–column pairing changes, so \(AB\neq BA\) for most matrices. Multiplication is, however, associative: \[ (AB)C = A(BC) \] whenever the products are defined.

Identity matrix and scalar factor

The identity matrix \(I_n\) is the multiplicative identity for square matrices: \(AI_n=A\) and \(I_nA=A\) whenever the sizes match. In this calculator you can set \(A\) or \(B\) to an identity matrix if it is square. This is useful for sanity checks: multiplying by identity should leave the other matrix unchanged.

A scalar factor can be applied to the product: \(k(AB)\). Because scalar multiplication distributes, you also have \(k(AB)=(kA)B=A(kB)\). The calculator includes an optional scalar \(k\) input to show how scaling affects the final matrix.

University note: fast multiplication

The straightforward algorithm for multiplying two \(n\times n\) matrices uses three nested loops and performs \(O(n^3)\) arithmetic operations. In more advanced courses, you may learn about faster asymptotic methods such as Strassen’s algorithm and other divide-and-conquer techniques that reduce the exponent below 3. In practical numerical computing, performance often comes from optimized block multiplication and cache-friendly implementations rather than asymptotically faster but complex algorithms. The row–column view remains the conceptual foundation even when the implementation changes.

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