Free Matrix Multiplication Calculator
Multiply matrices instantly with step-by-step solutions. Learn matrix multiplication rules, dot products, and linear transformations. Perfect for linear algebra students and professionals.
🧮 Matrix Multiplication Calculator
This free matrix multiplication calculator computes the product of two matrices with detailed step-by-step solutions. Enter your matrices below and click "Calculate" to see the result with complete working. The calculator validates dimension compatibility and shows each dot product computation.
📚 Quick Examples
📖 What is Matrix Multiplication?
Mathematical Definition
Matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix $A$ of dimensions $m \times n$ and matrix $B$ of dimensions $n \times p$, their product $AB$ is an $m \times p$ matrix.
Each element $c_{ij}$ of the product matrix is the dot product of the $i$-th row of $A$ and the $j$-th column of $B$. This operation is fundamental in linear algebra, computer graphics, physics, and data science.
Matrix Multiplication Rules
✅ Dimension Compatibility
Matrix $A$ ($m \times n$) can multiply matrix $B$ ($p \times q$) only if $n = p$. The result has dimensions $m \times q$.
🔄 Associative Property
$(AB)C = A(BC)$ when dimensions are compatible. The order of multiplication can be regrouped.
⚖️ Distributive Property
$A(B + C) = AB + AC$ and $(A + B)C = AC + BC$ when dimensions are compatible.
⚠️ Important Note: Non-Commutative
Matrix multiplication is NOT commutative in general: $AB \neq BA$. The order matters! This is different from regular number multiplication.
Step-by-Step Calculation Example
Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$
Step 1: Check dimensions: Both are 2×2, so product exists and is 2×2.
Step 2: Compute $c_{11}$ = (1×5) + (2×7) = 5 + 14 = 19
Step 3: Compute $c_{12}$ = (1×6) + (2×8) = 6 + 16 = 22
Step 4: Compute $c_{21}$ = (3×5) + (4×7) = 15 + 28 = 43
Step 5: Compute $c_{22}$ = (3×6) + (4×8) = 18 + 32 = 50
Result: $AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$
🚀 Real-World Applications
Where is Matrix Multiplication Used?
🤖 Computer Graphics
- 3D transformations (rotation, scaling, translation)
- Perspective projections
- Animation and game development
- Image processing filters
📈 Data Science & AI
- Neural network computations
- Principal Component Analysis (PCA)
- Linear regression models
- Recommendation systems
🔬 Physics & Engineering
- Quantum mechanics operations
- Circuit analysis
- Structural analysis
- Control systems
💼 Economics & Finance
- Portfolio optimization
- Input-output models
- Markov chains
- Risk analysis
⚠️ Common Mistakes & How to Avoid Them
❌ Mistake 1: Incorrect Dimension Compatibility
Trying to multiply matrices where columns of A ≠ rows of B.
Solution: Always check dimensions first. Matrix A (m×n) can only multiply B (p×q) if n = p.
❌ Mistake 2: Assuming Commutativity
Assuming AB = BA, which is generally false for matrices.
Solution: Remember matrix multiplication order matters. Test with simple examples to build intuition.
❌ Mistake 3: Incorrect Dot Product Calculation
Multiplying corresponding elements instead of row-column dot products.
Solution: Use our calculator's step-by-step feature to see exactly how each element is computed.
❓ Matrix Multiplication FAQ
Q: Can I multiply a 2×3 matrix by a 3×2 matrix?
Yes! A 2×3 matrix can multiply a 3×2 matrix because the inner dimensions match (3 and 3). The result will be a 2×2 matrix.
Q: What is the identity matrix in multiplication?
The identity matrix $I$ has 1's on the diagonal and 0's elsewhere. For any matrix $A$, $AI = IA = A$. It's the matrix equivalent of the number 1.
Q: How is matrix multiplication different from scalar multiplication?
Scalar multiplication multiplies every element by a single number. Matrix multiplication combines two matrices using dot products, resulting in a completely different operation with specific dimension rules.
Q: What is the zero matrix property?
The zero matrix $O$ has all elements equal to 0. For any compatible matrix $A$, $AO = O$ and $OA = O$, similar to multiplying by zero with regular numbers.
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