# Matrices I

• ### Introduction

A matrix (plural : matrices) is a rectangular arrangement of $m \times n$ numbers in $m$ rows and $n$ columns. It is written in between a pair of rectangular brackets. The numbers belonging to a matrix are known as its elements. Matrices are represented by capital Roman letters. For example, consider the following matrix $M$.

${M = \begin{bmatrix} 2& 3 & 0 \\[0.3em] 1 & -3 & \sqrt 2 \end{bmatrix}}$

It has 2 rows and 3 columns. Hence it is a $2 \times 3$ matrix.

If a matrix has $m$ rows and $n$ columns, it is said to be of the order $m$ by $n$. So, $M$ is of the order 2 by 3 and has 6 elements.

• ### Representation of an Element

$a_{ij}$ corresponds to that element of a matrix, which is present in the $i$th row and $j$th column of it. So, for matrix $M$,

${a_{21} = 1, a_{13}= 0, a_{23}= \sqrt 2}$

• ### Transpose of a matrix

A matrix obtained by interchanging the rows and columns of a matrix is known as the transpose of the matrix. So,

$M^T = \begin{bmatrix} 2& 1 \\[0.3em] 3 & -3 \\[0.3em] 0 & \sqrt 2 \end{bmatrix}$

• ### Equality of 2 matrices

2 matrices A and B are equal iff (if and only if)

i) their order is same and

ii) $\forall$ $a$ in $A$ and $b$ in $B$, $a_{ij}= b_{ij}$

• ### Row Matrix

A row matrix has only 1 row and more than 1 column. e.g.

${P = [1 \ 3 \ 2]}$

• ### Column Matrix

A column matrix has only 1 column and more than 1 row. e.g.

${Q = \begin {bmatrix} 2 \\ 3 \\ 5 \\ 6 \end {bmatrix}}$

• ### Null Matrix

A null matrix is the one, whose all elements are $0$. e.g.

${R = \begin {bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end {bmatrix}}$

• ### Square Matrix

A square matrix has equal number of rows and columns. So, the total number of elements in a square matrix is always a perfect square. e.g.

${S = \begin {bmatrix} 1 & 4 & \sqrt 5 & -6 \\ 77 & 64 & \frac 1 2 & 0 \\ e & \pi & 5 & 3 \\ 0 & 1 & 1 & 4 \end {bmatrix}}$

So, $S$ has 4 rows as well as 4 columns, so 16 elements.

The elements $a_{ij}$ of a square matrix, for which $i=j$ are known as the diagonal elements. Thus, $1,64, 5$ and $4$ are diagonal elements of $S$.

• ### Diagonal Matrix

A square matrix, whose all elements are zero, except the diagonal elements, is known as the diagonal matrix.e.g.

${D = \begin {bmatrix} 3 & 0 & 0 \\ 0 & -8 & 0 \\ 0 & 0 & 13 \end {bmatrix}}$

• ### Identity Matrix

An identity matrix is a square matrix, whose diagonal elements are equal to 1 and all other elements are 0. This matrix is of special importance in the matrix theory. Similar to the number $1$ in the theory of numbers, it acts as a unit entity. It is generally denoted by $I$. So, for any square matrix $A$, $AI = A$. Note that $A$ and $I$ must have the same order.

${I_{2} = \begin {bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}}$

${I_{3} = \begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {bmatrix}}$

• ### Upper- and Lower-Triangular Matrices

An upper-triangular matrix is the one, whose elements below the diagonal are $0$. Similarly, a lower-triangular matrix is the one, whose elements above the diagonal are $0$.

${U = \begin {bmatrix} 1 & 5 & 6 \\ 0 & 3 & 9 \\ 0 & 0 & 34 \end {bmatrix}}$

${L = \begin {bmatrix} 6 & 0 & 0 \\ 2 & 2 & 0 \\ 3 & 5 & 5 \end {bmatrix}}$