MATRIX:

A matrix is a rectangular array of numbers, each of which is called an entry of the matrix.

If a given matrix has $m$ rows and $n$ columns, it is called an $m\times n$ matrix, and the number $m$ and $n$ are the dimensions of the matrix

for example, in this matrix, it has 2 rows, and 3 columns, so the dimensions of this matrix would be

$2\times 3$

We sometimes refer to an $m\times 1$ matrix as a column vector and a $1\times n$ matrix as a row vector

the one on the left is referred to as a column vector since it has 1 column and $m$ rows, the one on the right is referred to as a row vector since it has 1 row and $n$ columns

A matrix is a square if it has the same number of rows as columns, In this case, the number of rows or columns is called the order of the matrix

The way we denoted matrices is by using capital letters, and the entries of a matrix are denoted by the corresponding lowercase letter with “double subscripts”. As an example, if we had this $3\times 3$ matrix:

If we wanted the value in the 2nd row 3rd column, instead of saying “the entry in the matrix A in the 2nd row 3rd column is…” you can just say $a_{23}=$, just makes naming easier

So, if we had this matrix:

and we wanted the value of the 1st row, 2nd column, we would write $b_{12}=2$. If we wanted the value of the 3rd row 1st column, this would be $b_{31}=0$

SUM OF MATRICIES:

Let $A$ and $B$ be matrices with the same dimensions. The sum of $A$ and $B$, written as $A+B$, is the matrix obtained by adding corresponding entries of $A$ and $B$. So, you would add $a_{11}+b_{11}, a_{12}+b_{12}$ and so on

Example:

Find the sum of these two matrices

MULTIPLICATION BY SCALAR:

Let $A$ be a matrix and $c$ be a scalar. Then, the scalar multiple $cA$ is the matrix obtained by multiplying each entry of $A$ by $c$

Example:

Find the value of $2A$

There is also something called a zero matrix but I am not making a header for it… can you GUESS what a zero matrix is…………………… this isn't rocket science

MATRIX MULTIPLICATION:

Let $A$ and $B$ be matrices such that the number of columns of $A$ is equal to the number of rows in $B$. Let $A$ be a $m\times n$ matrix, and $B$ be $n\times q$. Then, the product $AB$ is the $m\times q$ matrix, such that the entry in the $i^{th}$ row and $j^{th}$ column is the dot product of the $i^{th}$ row of $A$ with the $j^{th}$ column of $B$

• So, if you wanted to multiply a $2\times 3$ and $3\times 4$ matrix, this is fine, you end up with a $2\times 4$ matrix.
• However, if you wanted to multiply a $2\times 3$ and a $2\times 3$ matrix, you CAN’T, since the column of the first matrix is not equal to the row of the second matrix

It is also very important to know that $AB\neq BA$

The best way to explain this is through an example

Examples: Find the value of $AB$

Given the below values of $A$ and $B$, find the values of $AB$ and $BA$

DIAGONAL MATRIX:

The entries $a_{ii}$ of a square matrix $A$ form the main diagonal of $A$. For example, in the below matrix, the main diagonal of the matrix consists of 1, 2, 4

A square matrix in which every element NOT on the main diagonal is zero is called a diagonal matrix. It is important to know that it MUST be a square matrix

these are considered diagonal matrices, all the non main diagonal entries are 0

IDENTITY MATRIX:

An identity matrix, denoted by $I$, is a square matrix which has $1$’s all along the main diagonal, and zero everywhere else

If you multiply a non-zero matrix $A$ with an identity matrix, you are just going to get $A$ again

$AI=IA=A$