Matrix Multiplication Calculator
Verified Calculation Engine
The online Matrix Multiplication Calculator helps you calculate instantly and solve problems related to Matrices Determinants. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Engineering Math calculations.
Multiply two matrices when the columns of Matrix A equal the rows of Matrix B. Computes the product AB with step-by-step row–column sums.
(AB)_ij = Σ_k a_ik · b_kj
Matrix multiplication is valid when columns(A) = rows(B). If A is m×n and B is n×p, the product AB is m×p. Each entry is the dot product of a row of A with a column of B.
- Matrix A columns must equal Matrix B rows (inner dimensions match).
- Entries are real numbers (integers or decimals).
- Floating-point evaluation may introduce small rounding error for non-terminating decimals.
Inputs
Results
Worked Examples
Multiply two 2×2 matrices
Standard square product AB when both matrices are 2×2.
- matrix1: [ [ -1, 2 ], [ 3, -4 ] ]
- matrix2: [ [ 5, -2 ], [ -1, 6 ] ]
- result: [-7,14], [19,-30]
Multiply 2×3 by 3×2 matrices
Rectangular multiplication: columns of A match rows of B; result is 2×2.
- matrix1: [ [ 1, 2, 3 ], [ 4, 5, 6 ] ]
- matrix2: [ [ 7, 8 ], [ 9, 10 ], [ 11, 12 ] ]
- result: [58,64], [139,154]
Multiply matrices with decimals and negatives
Real-valued entries including decimals; product uses the same row–column rule.
- matrix1: [ [ 1.5, -2 ], [ 0.5, 3 ] ]
- matrix2: [ [ 2, 4 ], [ -1, 0.5 ] ]
- result: [5,5], [-2,3.5]
About this calculator
Overview
Matrix multiplication computes the product AB when the number of columns in Matrix A equals the number of rows in Matrix B. Each entry of the result is a row–column dot product: (AB)_ij = Σ_k a_ik · b_kj.
When to use
Use this matrix multiplication calculator for homework checks, transforming vectors/states, composing linear maps, or verifying AB when A is m×n and B is n×p. The result is an m×p matrix.
Inputs explained
- Matrix A — m×n numeric matrix (left factor)
- Matrix B — n×p numeric matrix (right factor; rows must equal columns of A)
Formula / method
Confirm columns(A) = rows(B). For each output position (i,j), compute the sum over k of a_ik × b_kj. Formula: (AB)_ij = Σ_k a_ik · b_kj. If A is m×n and B is n×p, C = AB is m×p. Note that AB and BA are generally different (and BA may be impossible).
Worked example
Example: A = [[1, 2, 3], [4, 5, 6]] (2×3) and B = [[7, 8], [9, 10], [11, 12]] (3×2). Then C11 = (1×7)+(2×9)+(3×11) = 58, C12 = (1×8)+(2×10)+(3×12) = 64, C21 = (4×7)+(5×9)+(6×11) = 139, C22 = (4×8)+(5×10)+(6×12) = 154, so AB = [[58, 64], [139, 154]].
Interpreting results
The Result Matrix is the product AB. Its dimensions are (rows of A) × (columns of B). Steps list each C_ij from the actual entered values.
Assumptions
- Columns of Matrix A equal rows of Matrix B.
- Entries are real numbers.
- Display precision may round intermediate floating-point values.
Limitations
- If columns(A) ≠ rows(B), multiplication is undefined — the calculator shows a validation message and no result matrix.
- This tool performs numeric matrix products; it is not a symbolic computer algebra system.
- Critical engineering decisions should be cross-checked by hand or with certified software.
Important note
Educational engineering-mathematics tool. Verify critical designs with hand analysis or certified software.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
Confirm columns(A) = rows(B). For each output position (i,j), compute the sum over k of a_ik × b_kj. Formula: (AB)_ij = Σ_k a_ik · b_kj. If A is m×n and B is n×p, C = AB is m×p. Note that AB and BA are generally different (and BA may be impossible).
Example Calculation
Example: A = [[1, 2, 3], [4, 5, 6]] (2×3) and B = [[7, 8], [9, 10], [11, 12]] (3×2). Then C11 = (1×7)+(2×9)+(3×11) = 58, C12 = (1×8)+(2×10)+(3×12) = 64, C21 = (4×7)+(5×9)+(6×11) = 139, C22 = (4×8)+(5×10)+(6×12) = 154, so AB = [[58, 64], [139, 154]].
Frequently Asked Questions
What is Matrix Multiplication Calculator?
- Matrix multiplication computes the product AB when the number of columns in Matrix A equals the number of rows in Matrix B. Each entry of the result is a row–column dot product: (AB)_ij = Σ_k a_ik · b_kj.
How does Matrix Multiplication Calculator work?
- Confirm columns(A) = rows(B). For each output position (i,j), compute the sum over k of a_ik × b_kj. Formula: (AB)_ij = Σ_k a_ik · b_kj. If A is m×n and B is n×p, C = AB is m×p. Note that AB and BA are generally different (and BA may be impossible).
Why use this Engineering Math calculator?
- Use this matrix multiplication calculator for homework checks, transforming vectors/states, composing linear maps, or verifying AB when A is m×n and B is n×p. The result is an m×p matrix.