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Chapter 8: Matrices

A matrix is an array of elements that follow certain rules:

$$A = \begin{bmatrix}1 & 6 & 3\\-1 & 2 & 4\end{bmatrix}$$

We use square brackets to enclose the array.

A matrix has $n$ rows and $m$ columns, and is said to be of order $n \times m$.

Always mention the rows first. Thus, the above example $A$ is $2 \times 3$

Examples

$$A = \begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}$$

is $3 \times 3$

$$B = \begin{bmatrix}1 & 1 & 0\end{bmatrix}$$

is $1 \times 3$

$$C = \begin{bmatrix}1\\-2\\9\\0\end{bmatrix}$$

is $4 \times 1$

$$D = \begin{bmatrix}1 & 0 & 0 & 2\\0 & 1 & 0 & 4\\0 & 0 & 1 & 5\\0 & 0 & 0 & 1\end{bmatrix}$$

is $4 \times 4$

A matrix is square if the # of rows == # of columns.

Some common square matrices:

• zero matrix is all 0s
• identity matrix is all 0s except along the main diagonal (top left to bottom right) is 1s

In computer graphics, we most commonly use square $2 \times 2$, $3 \times 3$, and $4 \times 4$ matrices.

Operations

Two matrices are equal if

• they have the same # of rows and columns
• their corresponding elements in each position are equal

$$A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$$ $$B = \begin{bmatrix}1 & x\\y & 4\end{bmatrix}$$

If $A = B$, then $x = 2$ and $y = 3$.

Addition and Subtraction

Addition and subtraction of matrices is defined by addition and subtraction of corresponding elements.

We can only add and subtract matrices of the same order.

$$\begin{bmatrix}2 & 1 & 3\\4 & 1 & -1\end{bmatrix} + \begin{bmatrix}3 & 2 & 0\\0 & 1 & 0\end{bmatrix}$$ $$= \begin{bmatrix}2 + 3 & 1 + 2 & 3 + 0\\4 + 0 & 1 + 1 & -1 + 0\end{bmatrix} = \begin{bmatrix}5 & 3 & 3\\4 & 2 & -1\end{bmatrix}$$

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