matrices

What is matrix?

matrix are list / array of number / functions that are represented with enclosed brackets

what is linear transformations

Operations on Matrices

Addition / subtraction

Multiplication (scalar)

Multiplication

More information about matrices

Order of matrices

\begin{bmatrix} 1 & 2 & 3 \\ 2 & n.. & n.. \\ 3 & n.. & n.. \end{bmatrix}

order of matrix is given by $m \times n$ . where $m$ is number of rows and $n$ is number of columns. for above matrix it is $3 \times 3 = 9$ as $m,n=3$ .

Two matrices are considered equal if both have same order and there components are same

\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

Types of matrices

Column

\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ \end{bmatrix}

Row

\begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \end{bmatrix}

square

\begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & n.. & n.. & n.. & n.. & n.. \\ 3 & n.. & n.. & n.. & n.. & n.. \\ 4 & n.. & n.. & n.. & n.. & n.. \\ 5 & n.. & n.. & n.. & n.. & n.. \\ 6 & n.. & n.. & n.. & n.. & n.. \\ \end{bmatrix}

Diagonal

\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 \\ \end{bmatrix}

In this type of Matrix everything accept diagonal values are zero

Scalar Matrix

\begin{bmatrix} 5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 & 0 & 0 \\ 0 & 0 & 5 & 0 & 0 & 0 \\ 0 & 0 & 0 & 5 & 0 & 0 \\ 0 & 0 & 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5 \\ \end{bmatrix}

Diagonal Matrix with every diagonal value being same

Identify Matrix

\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \Huge , \small \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Scalar Matrix with every diagonal value being 1 Identify Matrix has use in Transformations it’s basics building block in many mathematical systems

Zero

\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}