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Matrices and Matrix Operations
Matrices and Matrix Operations
A matrix is a rectangular array of numbers used to organize coefficients, transformations, and data.
Notation
A matrix with $m$ rows and $n$ columns is an $m \times n$ matrix.
Basic Operations
- Add and subtract matrices entry-by-entry.
- Multiply a matrix by a scalar by multiplying each entry.
- Matrix multiplication is defined when inner dimensions match.
Example 1
Add $$\begin{bmatrix}1&2\3&4\end{bmatrix}+\begin{bmatrix}5&6\7&8\end{bmatrix}.$$
$$\begin{bmatrix}6&8\10&12\end{bmatrix}$$
Example 2
Compute $$2\begin{bmatrix}3&-1\0&5\end{bmatrix}.$$
$$\begin{bmatrix}6&-2\0&10\end{bmatrix}$$
Example 3
Multiply $$\begin{bmatrix}1&2\0&1\end{bmatrix}\begin{bmatrix}3\4\end{bmatrix}.$$
$$\begin{bmatrix}11\4\end{bmatrix}$$
Practice Problems
1. Add $$\begin{bmatrix}2&1\0&3\end{bmatrix}+\begin{bmatrix}4&-1\5&2\end{bmatrix}.$$
2. Compute $$-3\begin{bmatrix}1&-2\4&0\end{bmatrix}.$$
3. State the size of a $3 \times 2$ matrix.
4. When is matrix multiplication defined?
5. Multiply $$\begin{bmatrix}2&0\1&3\end{bmatrix}\begin{bmatrix}1\2\end{bmatrix}.$$
Show Answer Key
1. $$\begin{bmatrix}6&0\5&5\end{bmatrix}$$
2. $$\begin{bmatrix}-3&6\-12&0\end{bmatrix}$$
3. 3 rows, 2 columns
4. When the number of columns of the first equals the number of rows of the second
5. $$\begin{bmatrix}2\7\end{bmatrix}$$