Overview
This lecture explains how to multiply matrices, determine when the operation is defined, and how to compute the resulting matrix and its size.
Matrix Multiplication Basics
- Matrix multiplication order matters: ( AB \neq BA ) in general.
- To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second.
- The size (order) of the product matrix is: (rows of first) × (columns of second).
Example 1: Multiplying 1×3 and 3×1 Matrices
- Matrix ( A ) is 1×3 (one row, three columns); matrix ( B ) is 3×1 (three rows, one column).
- ( AB ) results in a 1×1 matrix: multiply each element pair, sum the products.
- Calculation: ( 2×3 + 5×4 + 6×(-5) = 6+20-30 = -4 ); ( AB = [-4] ).
- ( BA ) results in a 3×3 matrix: each entry is row of ( B ) times column of ( A ).
- ( BA ) entries: ( \begin{bmatrix} 6 & 15 & 18 \ 8 & 20 & 24 \ -10 & -25 & -30 \end{bmatrix} ).
Example 2: Multiplying 2×3 and 3×4 Matrices
- Matrix ( A ) is 2×3; matrix ( B ) is 3×4.
- ( AB ) is defined, resulting in a 2×4 matrix.
- ( BA ) cannot be computed (incompatible sizes: 4 columns in ( B ), 2 rows in ( A )).
- To find each entry, multiply row ( i ) of ( A ) by column ( j ) of ( B ), add products.
- The computed ( AB ) matrix:
[
\begin{bmatrix}
21 & 8 & 10 & 20 \
42 & -18 & 2 & 40
\end{bmatrix}
]
Key Terms & Definitions
- Order of a Matrix — Number of rows × number of columns in the matrix.
- Matrix Multiplication — An operation where each entry of the product is the sum of products of elements from a row in the first matrix and a column in the second matrix.
- Defined Multiplication — Matrix ( AB ) is defined if columns of ( A ) = rows of ( B ).
Action Items / Next Steps
- Practice multiplying matrices with different sizes and verify the resulting orders.
- Review how to compute individual entries by multiplying rows by columns.
- Make sure to always check matrix sizes before attempting to multiply.