Conditions to Multiply Matrices

Matrix Multiplication Rules - When Can We Multiply Matrices?

1️⃣ Learn the fundamental rule for matrix multiplication compatibility

2️⃣ Understand how inner dimensions must match for multiplication

3️⃣ Master predicting result matrix dimensions before calculating

4️⃣ Identify which matrices can and cannot be multiplied

5️⃣ Practice with real examples of dimension checking

🔍 The Golden Rule of Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. If they are not equal, we cannot multiply them.

Matrix Multiplication Rule

\text{Columns of Matrix A} = \text{Rows of Matrix B}

Inner dimensions must match for multiplication to be possible

📐 Understanding Matrix Dimensions

When we have a 3×2 matrix (3 rows, 2 columns), it can only be multiplied by matrices that have exactly 2 rows. Let's see why this matters.

3×2 Matrix Multiplication Examples

✅ Valid: 3×2 × 2×3

\underbrace{\begin{bmatrix} \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \end{bmatrix}}_{3×2} \times \underbrace{\begin{bmatrix} \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \end{bmatrix}}_{2×3} = \text{Possible!}

Columns of first matrix (2) = Rows of second matrix (2) ✓

✅ Valid: 3×2 × 2×4

\underbrace{\begin{bmatrix} \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \end{bmatrix}}_{3×2} \times \underbrace{\begin{bmatrix} \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \end{bmatrix}}_{2×4} = \text{Possible!}

As long as the second matrix has 2 rows, multiplication works

❌ Invalid: 3×2 × 3×2

\underbrace{\begin{bmatrix} \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \end{bmatrix}}_{3×2} \times \underbrace{\begin{bmatrix} \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \end{bmatrix}}_{3×2} = \text{Impossible!}

Columns of first matrix (2) ≠ Rows of second matrix (3) ❌

🎯 Predicting Result Dimensions

Before we even start calculating, we can predict the dimensions of the result matrix. The result takes the outer dimensions from the multiplication.

Result Dimension Formula

(m × n) \times (n × p) = (m × p)

Outer dimensions give result size, inner dimensions must match

Step-by-Step Dimension Analysis

\underbrace{(3×2)}_{Matrix\,A} \times \underbrace{(2×4)}_{Matrix\,B} = \underbrace{(3×4)}_{Result}

🔍 Analysis Breakdown

First number (3): Represents rows of the result matrix
Last number (4): Represents columns of the result matrix
Inner numbers (2, 2): Must be equal for multiplication to work
Result prediction: We know it will be a 3×4 matrix before calculating

✅ Matrix Compatibility Check Process

Always follow this systematic approach to check if two matrices can be multiplied and predict the result dimensions.

Matrix Multiplication Checklist

Step 1: Identify Matrix Dimensions

Write down the dimensions of both matrices: Matrix A (rows × columns), Matrix B (rows × columns)

Step 2: Check Inner Dimensions

Compare: Columns of Matrix A = Rows of Matrix B? If yes, multiplication is possible.

Step 3: Predict Result Dimensions

Result matrix will have: (Rows of Matrix A) × (Columns of Matrix B)

Step 4: Set Up Framework

Before calculating, draw the result matrix framework with correct dimensions

🧮 Practical Examples

Multiple Multiplication Examples

Example 1: (2×3) × (3×1)

Check: Columns of first (3) = Rows of second (3) ✓
Result: (2×1) matrix - a column vector with 2 elements

Example 2: (4×2) × (2×5)

Check: Columns of first (2) = Rows of second (2) ✓
Result: (4×5) matrix with 20 elements to calculate

Example 3: (3×4) × (2×3)

Check: Columns of first (4) ≠ Rows of second (2) ❌
Result: Multiplication is impossible

Example 4: (1×5) × (5×1)

Check: Columns of first (5) = Rows of second (5) ✓
Result: (1×1) matrix - a single number (scalar)

🧠 Key Rules to Remember

Essential Multiplication Rules

Compatibility Rule: Columns of first matrix = Rows of second matrix
Result Prediction: (m×n) × (n×p) = (m×p)
Inner Dimensions: Must match exactly for multiplication to work
Outer Dimensions: Determine the size of the result matrix
Always Check First: Verify compatibility before calculating

🎯 Matrix Multiplication Fundamentals

Golden Rule: Columns of Matrix A must equal rows of Matrix B
Dimension Check: Always verify compatibility before attempting multiplication
Result Size: Outer dimensions determine the result matrix size
Systematic Approach: Write dimensions, check compatibility, predict result
Common Mistake: Don't assume all matrices can be multiplied
Preview Success: Know your result dimensions before calculating