Matrix Multiplication

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Matrix Multiplication Process - Step-by-Step Method

1️⃣ Learn the systematic approach to multiplying matrices
2️⃣ Master the row × column multiplication technique
3️⃣ Understand how to fill result matrix positions systematically
4️⃣ Practice element-wise multiplication and summing
5️⃣ Complete full matrix multiplication examples

🚀 Starting the Multiplication Process

Let's multiply a 2×3 matrix by a 3×4 matrix. First, we verify the inner dimensions match (3 = 3 ✓) and predict our result will be a 2×4 matrix.

Matrix Multiplication Setup

\underbrace{\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}}_{2×3} \times \underbrace{\begin{bmatrix} b_{11} & b_{12} & b_{13} & b_{14} \\ b_{21} & b_{22} & b_{23} & b_{24} \\ b_{31} & b_{32} & b_{33} & b_{34} \end{bmatrix}}_{3×4} = \underbrace{\begin{bmatrix} \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \end{bmatrix}}_{2×4}
Result matrix: 2 rows and 4 columns = 8 positions to fill

🔄 Step 1: Fill First Row of Result Matrix

Take the first row from the first matrix and multiply it by each column of the second matrix to fill the first row of the result.

First Row Calculation Process

Position (1,1): Row 1 × Column 1

[a_{11}, a_{12}, a_{13}] \times \begin{bmatrix} b_{11} \\ b_{21} \\ b_{31} \end{bmatrix} = a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31}

First row × First column → Place result in position (1,1)

Position (1,2): Row 1 × Column 2

[a_{11}, a_{12}, a_{13}] \times \begin{bmatrix} b_{12} \\ b_{22} \\ b_{32} \end{bmatrix} = a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32}

First row × Second column → Place result in position (1,2)

Position (1,3): Row 1 × Column 3

[a_{11}, a_{12}, a_{13}] \times \begin{bmatrix} b_{13} \\ b_{23} \\ b_{33} \end{bmatrix} = a_{11}b_{13} + a_{12}b_{23} + a_{13}b_{33}

First row × Third column → Place result in position (1,3)

Position (1,4): Row 1 × Column 4

[a_{11}, a_{12}, a_{13}] \times \begin{bmatrix} b_{14} \\ b_{24} \\ b_{34} \end{bmatrix} = a_{11}b_{14} + a_{12}b_{24} + a_{13}b_{34}

First row × Fourth column → Place result in position (1,4)

First row complete:
\begin{bmatrix} \text{result}_{11} & \text{result}_{12} & \text{result}_{13} & \text{result}_{14} \\ \_ & \_ & \_ & \_ \end{bmatrix}

🔄 Step 2: Fill Second Row of Result Matrix

Now take the second row from the first matrix and multiply it by each column of the second matrix to complete the result matrix.

Second Row Calculation Process

Position (2,1): Row 2 × Column 1

[a_{21}, a_{22}, a_{23}] \times \begin{bmatrix} b_{11} \\ b_{21} \\ b_{31} \end{bmatrix} = a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31}

Second row × First column → Place result in position (2,1)

Position (2,2): Row 2 × Column 2

[a_{21}, a_{22}, a_{23}] \times \begin{bmatrix} b_{12} \\ b_{22} \\ b_{32} \end{bmatrix} = a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32}

Second row × Second column → Place result in position (2,2)

Position (2,3): Row 2 × Column 3

[a_{21}, a_{22}, a_{23}] \times \begin{bmatrix} b_{13} \\ b_{23} \\ b_{33} \end{bmatrix} = a_{21}b_{13} + a_{22}b_{23} + a_{23}b_{33}

Second row × Third column → Place result in position (2,3)

Position (2,4): Row 2 × Column 4

[a_{21}, a_{22}, a_{23}] \times \begin{bmatrix} b_{14} \\ b_{24} \\ b_{34} \end{bmatrix} = a_{21}b_{14} + a_{22}b_{24} + a_{23}b_{34}

Second row × Fourth column → Place result in position (2,4)

🎯 The Row × Column Technique Explained

The key to matrix multiplication is understanding how to multiply a row by a column. This involves element-wise multiplication followed by addition.

Row × Column Formula

[r_1, r_2, r_3] \times \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = r_1 \cdot c_1 + r_2 \cdot c_2 + r_3 \cdot c_3
Multiply corresponding elements, then add all products
Step-by-Step Row × Column Process

Step 1: Align Elements

Take the first element from the row and multiply by the first element of the column

r_1 \times c_1

Step 2: Continue Element-wise

Take the second element from the row and multiply by the second element of the column

r_2 \times c_2

Step 3: Complete All Pairs

Continue until all corresponding elements are multiplied

r_3 \times c_3, etc.

Step 4: Sum All Products

Add all the products together to get the final result for this position

r_1c_1 + r_2c_2 + r_3c_3 + ...

✅ Complete Matrix Multiplication Result

After completing all row × column calculations, we get our final result: a 2×4 matrix with 2 rows and 4 columns, exactly as predicted.

Final Result Matrix

\begin{bmatrix} a_{11}b_{11}+a_{12}b_{21}+a_{13}b_{31} & a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32} & a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33} & a_{11}b_{14}+a_{12}b_{24}+a_{13}b_{34} \\ a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31} & a_{21}b_{12}+a_{22}b_{22}+a_{23}b_{32} & a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33} & a_{21}b_{14}+a_{22}b_{24}+a_{23}b_{34} \end{bmatrix}
A complete 2×4 matrix as predicted from dimension analysis!

🔢 Worked Numerical Example

Complete Numerical Example
\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 2 & 1 \\ 2 & 1 & 0 & 3 \\ 0 & 3 & 1 & 2 \end{bmatrix}

First Row Results

(1,1): 1×1 + 2×2 + 3×0 = 5
(1,2): 1×0 + 2×1 + 3×3 = 11
(1,3): 1×2 + 2×0 + 3×1 = 5
(1,4): 1×1 + 2×3 + 3×2 = 13

Second Row Results

(2,1): 4×1 + 5×2 + 6×0 = 14
(2,2): 4×0 + 5×1 + 6×3 = 23
(2,3): 4×2 + 5×0 + 6×1 = 14
(2,4): 4×1 + 5×3 + 6×2 = 31
Final Answer:
\begin{bmatrix} 5 & 11 & 5 & 13 \\ 14 & 23 & 14 & 31 \end{bmatrix}

🧠 Multiplication Process Summary

Step-by-Step Process

  1. Verify Compatibility - Check inner dimensions match
  2. Predict Result Size - Use outer dimensions
  3. Set Up Framework - Draw empty result matrix
  4. Fill Row by Row - Take each row from first matrix
  5. Multiply by Each Column - Row × each column of second matrix
  6. Element-wise Multiply & Sum - Corresponding elements, then add

🎯 Key Multiplication Process Rules

  • Systematic Approach: Work row by row from the first matrix
  • Row × Column Rule: Each position = one row × one column
  • Element-wise Multiplication: Multiply corresponding elements, then sum
  • Position Tracking: Result position (i,j) = Row i × Column j
  • Complete Coverage: Every row must multiply every column
  • Final Check: Result matrix should have predicted dimensions
20. Conditions to Multiply Matrices22. Determinant of Matrices 2x2
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