Precalculus / Unit 8: Systems and Matrices
8.2 Matrix Operations
Notes
A matrix is a rectangular array of numbers. An $m \times n$ matrix has $m$ rows and $n$ columns.
- Add or subtract matrices of the same size entry by entry. A scalar multiplies every entry.
- Multiplying $A$ ($m \times n$) by $B$ ($n \times p$) is possible only if the inner dimensions match; the result is $m \times p$.
- Each entry of $AB$ is a row of $A$ dotted with a column of $B$.
- Matrix multiplication is not commutative: usually $AB \ne BA$.
- Example: for $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$.
Practice
A is $2 \times 3$ and B is $3 \times 4$. What is the size of $AB$?
The inner dimensions (3 and 3) match, and the result has the outer dimensions.
Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ -1 & 5 \end{bmatrix}$. What is $AB$?
Row 1: $1(0) + 2(-1) = -2$, $1(1) + 2(5) = 11$. Row 2: $3(0) + 4(-1) = -4$, $3(1) + 4(5) = 23$.
Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ -1 & 5 \end{bmatrix}$. What is $3A - B$?
$3A = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix}$; subtract $B$ entry by entry.
Ready for more? Take the Unit 8 test.