Definition of a Matrix

A matrix is a list of numbers arranged into rows and columns. They are usually represented with capital letters and they are drawn as arrays of numbers in square brackets. For example, $$A = \begin{bmatrix} 1 & 3 & -7 \\ 2 & 0 & 0 \\ 3 & -1 & 1 \end{bmatrix}$$

The rows go across and the columns go down. The second row of $A$ is the vector $<2, 0, 0>$ and the third column of $A$ is $$\begin{bmatrix} -7 \\ 0 \\ 1\end{bmatrix}$$

Specific entries are denoted with subscripts. The element in the first row and second column is $$A_{1,2} = 3$$ Keep in mind that the row always comes first and the column second.

The dimensions of a matrix are given as rows $\times$ columns. So the dimensions of $A$ are $2 \times 2.$

Matrix Addition

If $A$ and $B$ are two matrices of the same dimension, then they can be added coordinate-wise.

Example 1: $$\begin{bmatrix} 2 & 1 & 3 \\ 0 & 4 & 2 \end{bmatrix} + \begin{bmatrix} 7 & -2 & 0 \\ 1 & 1 & 5 \end{bmatrix} = \begin{bmatrix} 2+7 & 1+(-2) & 3+0 \\ 0+1 & 4+1 & 2+5 \end{bmatrix} = \begin{bmatrix} 9 & -1 & 3 \\ 1 & 5 & 7 \end{bmatrix}$$ Example 2: $$\begin{bmatrix} 0 & 5 & 0 \\ -3 & 1 & 1 \end{bmatrix} + \begin{bmatrix} 2 & 8 \\ 0 & -3 \end{bmatrix}$$ This sum does not exist the because the first matrix has dimensions $2 \times 3$ but the second matrix has dimensions $2 \times 2.$

Properties of the Matrix Addition

• Commutativity: If $A$ and $B$ have the same dimensions, then $$A+B=B+A.$$
• Identity: Given a matrix $A$ with dimensions $n \times m,$ there is a zero matrix $0$ of dimensions $n \times m$ such that $A + 0 = 0 + A = 0.$ The zero matrix has $0$ in every entry.
• Associativity: If $A,$ $B$ and $C$ are matrices that all have the same dimensions, then $(A+B)+C = A+(B+C).$

Scalar Multiplication

Scalar multiplication works in a similar way for matrices as it does for vectors. When multiplying a matrix by a number, every entry gets multiplied by the number.

Example: Let $$A = \begin{bmatrix} 2 & 3 \\ 0 & 1 \\ -2 & 4 \end{bmatrix}$$ Then $$3A = 3 \begin{bmatrix} 2 & 3 \\ 0 & 1 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 3*2 & 3*3 \\ 3*0 & 3*1 \\ 3*-2 & 3*4 \end{bmatrix} = \begin{bmatrix} 6 & 9 \\ 0 & 3 \\ -6 & 12 \end{bmatrix}$$

Matrix Subtraction

Subtracting matrices is a combination of scalar multiplication and matrix addition. If $A$ and $B$ are matrices of the same dimensions, $A - B = A + (-1)B.$

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