#### Addition of Matrices

Matrices add element to element in a fairly intuitive manner. Maybe best shown by example:

(1)
\begin{align} {\begin{pmatrix} a & b\\ c & d \\ \end{pmatrix}}+ {\begin{pmatrix} e & f \\ g & h \end{pmatrix}}= \begin{pmatrix} a+e & b+f \\ c+g & h+h \end{pmatrix} \end{align}

#### Subtraction of Matrices

Subtraction works the same as addition, that is element to element:

(2)
\begin{align} {\begin{pmatrix} a & b\\ c & d \\ \end{pmatrix}} - {\begin{pmatrix} e & f \\ g & h \end{pmatrix}} = {\begin{pmatrix} a-e & b-f \\ c-g & h-h \end{pmatrix}} \end{align}

#### Multiplication by a Scalar

A vector can by multiplied by a scalar (a regular number). In this case each element of the matrix is multiplied by the same value. For example:

(3)
\begin{align} 3{\begin{pmatrix} a & b\\ c & d \\ \end{pmatrix}} = {\begin{pmatrix} 3a & 3b\\ 3c &3d \\ \end{pmatrix}} \end{align}

### Matrix Equality

If two matrices are set equal to each other than each element is equal to the corresponding element. For example:

(4)
\begin{align} {\begin{pmatrix} 3 & x\\ 3y & 1 \\ \end{pmatrix}} = {\begin{pmatrix} y & 5\\ 9 & x-4 \\ \end{pmatrix}} \end{align}

From this matrix we can write 4 equations (this begins to show the usefulness and compactness of matrix notation).

(5)
\begin{equation} 3=y \end{equation}

(6)
\begin{equation} x = 5 \end{equation}

(7)
\begin{equation} 3y=9 \end{equation}

(8)
\begin{equation} 1 = x- 4 \end{equation}

These equations can be solved to show (if I wrote them correctly) that $y = 3$ and $x = 5$.

##### Example

Find *x* and *y* if:

(9)
\begin{align} {\begin{pmatrix} x & x^2\\ 3 & -1 \\ \end{pmatrix}} = {\begin{pmatrix} y & 4\\ 3 & y+1 \\ \end{pmatrix}} \end{align}

Again, we can write four equations from this matrix. We do this by setting corresponding elements equal to each other.

(10)
\begin{equation} x = y \end{equation}

(11)
\begin{equation} x^2 = 4 \end{equation}

(12)
\begin{equation} 3 = 3 \end{equation}

(13)
\begin{equation} -1 = y +1 \end{equation}

The last one turns out to be the most straight forward (and useful) showing that $y=-2$ which shows that $x = -2$ which agrees with the second equation $(-2)^2=4$.

#### Matrix Multiplication

Here is were things get a bit less clear, but this is also where we get the first tools that help use do what we couldn't (easily) do before, but the price is a bit of confusion at first. For example the multiplication of a 2x2 with another 2x2.

(14)
\begin{align} {\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}} {\begin{pmatrix} e & f \\ g&h \\ \end{pmatrix}} = {\begin{pmatrix} ae+bg & af+bh\\ ce+dg& cf+dh \end{pmatrix}} \end{align}

This is really best shown in person (where a teacher or fellow student can wave their hands), but I will make an attempt to explain… Notice that the element in the first row and first column is a product of the first row of the first matrix and the first column of the second matrix. Likewise the element in the second row and second column is a product of the second row of the first matrix and the second column of the second matrix.

The mechanics of matrix multiplication are somewhat arbitrary, in that humans simply decided on a consistent method, however some justification can be made from the idea that matrices are used to store information… (I'm working on my justification, will link when its done.

There are limits to matrix multiplication, in that not just any two matrices can be multiplied. An important fact of matrix multiplication is that in general:

(15)
\begin{align} A B \neq B A \end{align}

You can find special cases, but they are rare. In order for matrix multiplication to be defined the first matrix must have the same number of columns as the the second matrix has rows. For example :

(16)
\begin{align} {\begin{pmatrix} 1 & 2& 3\\ 4& 5 &6 \end{pmatrix}} {\begin{pmatrix} 1 & 2\\ 3 & 4\\ 5 & 6 \end{pmatrix}}= \begin{pmatrix} 22& 28 \\ 49& 64\\ \end{pmatrix} \end{align}

Notice that the order of the product is reduced to *2 x 2*. **In general if an ***m x n* order matrix is multiplied by an *n x k* order matrix the result will be an *m x k* matrix.

#### Identity Matrix

This one is pretty straight forward. An identity matrix is a matrix that has 1's on the diagonal and zeros everywhere else…

(17)
\begin{align} I = {\begin{pmatrix} 1 & 0& 0 &0 \\ 0& 1 &0 &0 \\ 0& 0 & 1 & 0\\ 0& 0 & 0 & 1 \end{pmatrix}} \end{align}

The identity matrix no matter the order is given the symbol of *I*.

The identity matrix can also be defined as the matrix that when mulitplied by another matrix, say **A** leaves **A** unchanged:

(18)
\begin{equation} A I = A \end{equation}

This is a bit more useful than it sounds, but not a ton…

#### Zero Matrix

This one is on the IB syllabus, so I'll mention it, but it's a bit silly. The zero matrix is a matrix with all zeros… For example:

(19)
\begin{align} {\begin{pmatrix} 0 & 0& 0 &0 \\ 0& 0 &0 &0 \\ 0& 0 & 0 & 0\\ 0& 0 & 0 & 0 \end{pmatrix}} \end{align}

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