1. What Is Matrix Multiplication?
Matrix multiplication is a method of multiplying two matrices to get a new matrix. Unlike addition or subtraction, multiplication is not done entry-by-entry.
Instead, multiplication uses the row-by-column rule: each element of the product is formed by multiplying a row from the first matrix with a column from the second matrix.
2. Condition for Matrix Multiplication
You can multiply two matrices only if:
\( A_{m \times n} \times B_{p \times q} \)
The number of columns of the first matrix must equal the number of rows of the second matrix:
\( n = p \)
- If this condition is not satisfied, multiplication is not defined.
The resulting matrix will have order:
\( m \times q \)
3. Rule of Matrix Multiplication (Row-by-Column)
If \( A = [a_{ij}] \) of order \( m \times n \) and \( B = [b_{ij}] \) of order \( n \times p \), then the product matrix \( AB = C = [c_{ij}] \) is obtained using:
\( c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} \)
This means:
- Take the i-th row of matrix A
- Take the j-th column of matrix B
- Multiply corresponding entries and add them up
4. Example 1: Multiplying a 2 × 3 Matrix with a 3 × 2 Matrix
Let:
\( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} \)
Here:
- A is 2 × 3
- B is 3 × 2
Since the inner numbers match (3 = 3), multiplication is possible.
4.1. Finding the (1,1) Element
1·7 + 2·9 + 3·11 = 7 + 18 + 33 = 58
4.2. Finding the (1,2) Element
1·8 + 2·10 + 3·12 = 8 + 20 + 36 = 64
4.3. Finding the (2,1) Element
4·7 + 5·9 + 6·11 = 28 + 45 + 66 = 139
4.4. Finding the (2,2) Element
4·8 + 5·10 + 6·12 = 32 + 50 + 72 = 154
4.5. Final Product Matrix
\( AB = \begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix} \)
5. Example 2: Multiplying Square Matrices
Let:
\( X = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}, \quad Y = \begin{bmatrix} 1 & 5 \\ 0 & 2 \end{bmatrix} \)
Both are 2 × 2 matrices.
5.1. Computing the Product
- \( (1,1): 2·1 + (-1)·0 = 2 \)
- \( (1,2): 2·5 + (-1)·2 = 10 - 2 = 8 \)
- \( (2,1): 3·1 + 4·0 = 3 \)
- \( (2,2): 3·5 + 4·2 = 15 + 8 = 23 \)
5.2. Final Answer
\( XY = \begin{bmatrix} 2 & 8 \\ 3 & 23 \end{bmatrix} \)
6. Important Observations About Matrix Multiplication
Matrix multiplication has special rules:
- Not commutative: \( AB \neq BA \) in general.
- Associative: \( A(BC) = (AB)C \)
- Distributive: \( A(B + C) = AB + AC \)
- Identity matrix: \( AI = IA = A \)
Understanding these properties is essential for solving complex matrix expressions.