Matrices

Math Algebra

A matrix is an $n$ by $m$ set of values. A $4\times 3$can be notated by:

$$\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\\ d_1& d_2& d_3\end{array}\right]$$

To get a value from matrix $a$ in row $r$ and column $c$, use:

$$a_{r,c}$$

Addition

With two matrices of the same order, add corresponding elements:

$$\left[\begin{array}{cc}{a}_1& b_1\\ c_1& d_1\end{array}\right]+\left[\begin{array}{cc}{a}_2& b_2\\ c_2& d_2\end{array}\right]=\left[\begin{array}{cc}{a}_1+a_2& b_1+b_2\\ c_1+c_2& d_1+d_2\end{array}\right]$$

Subtraction

With two matrices of the same order, subtract corresponding elements:

$$\left[\begin{array}{cc}{a}_1& b_1\\ c_1& d_1\end{array}\right]-\left[\begin{array}{cc}{a}_2& b_2\\ c_2& d_2\end{array}\right]=\left[\begin{array}{cc}{a}_1-a_2& b_1-b_2\\ c_1-c_2& d_1-d_2\end{array}\right]$$

Scalar Multiplication

When multiplying a matrix by a scalar, multiply each element by said scalar:

$$s\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}sa& sb\\ sc& sd\end{array}\right]$$

Matrix Multiplication

Let $a$ be an $i$ by $j$ matrix and $b$ be a $m$ by $n$ matrix. If $j=m$, $ab$ is defined.

$$a{b}_{c,d}=\sum _{k=1}^j{a}_{c,k}{b}_{k,d}$$