## Concept, Size and Types of Matrices

#### Overview

A rectangular array of numbers arranged in rows (horizontal lines) and columns (vertical lines) enclosed between round (or square) brackets is called a matrix. The size of the matrix or the order of the matrix is defined as the number of rows followed by the number of columns.

### Matrix

A rectangular array of numbers arranged in rows (horizontal lines) and columns (vertical lines) enclosed between round (or square) brackets is called a matrix. Here are some examples of matrices

$$i) \begin{pmatrix} 1 & 2& 3\\ 4 &5 & 6 \end{pmatrix}$$

$$ii)\begin{pmatrix} 7& 8\\ 9&5\\ 9 &0\end{pmatrix}$$

#### Size of Matrix

The size of the matrix or the order of the matrix is defined as the number of rows followed by the number of columns. Example,

i) is said to be of order$$2\times3$$ as there are two rows and three columns .

ii) is said to be of order $$3\times2$$ as there are three rows and two columns.

#### Types of Matrices

There are different types of matrices like row matrix,column matrix,square matrix, etc. They are described below.

i) Row matrix

A matrix which has only one row is called a row matrix. For examples, (1,2), (3,4,-1) etc .

ii) Column matrix

A matrix which has only one column is called a column matrix. For example,

$$\begin{pmatrix} 1 \\ 2 \end{pmatrix}and \begin{pmatrix} 3 \\ 4\\-2 \end{pmatrix}$$

iii) Square matrix

A matrix having the same number of row and columns is called a square matrix. For example,

$$\begin{pmatrix} 1 & 2\\ 4 &5 \end{pmatrix}and\begin{pmatrix} 1 & 2&3\\ 4 &5&6\\7&8&9 \end{pmatrix}$$

In example first, there are two rows and two columns and In example second, there are three rows and three columns .

iv) Rectangular Matrix

A matrix for which horizontal and vertical dimensions are not the same is called a rectangular matrix. For example,

$$\begin{pmatrix} 1 & 0& 3\\ -1 &0 & 6 \end{pmatrix}and\begin{pmatrix} 2& 8\\ 4&5\\ 2 &0\end{pmatrix}$$

v) Diagonal Matrix

A square matrix which has zeros everywhere other than the main diagonal is called a diagonal matrix. For example,

$$\begin{pmatrix} 1 & 0\\ 0 &5 \end{pmatrix}and\begin{pmatrix} 1 & 0&0\\ 0 &2&0\\0&0&3 \end{pmatrix}$$

vi) Scalar Matrix

A dialgonal matrix having all the diagonal elements equal is called a scalar matrix. For example,

$$\begin{pmatrix} 2 & 0\\ 0 &2 \end{pmatrix}and\begin{pmatrix} -6& 0&0\\ 0 &-6&0\\0&0&-6 \end{pmatrix}$$

vii) Unit Matrix or Identity Matrix

A diagonal matrix having all the diagonal elements equal to 1 is called a unit matrix. For example,

$$\begin{pmatrix} 1 & 0\\ 0 &1 \end{pmatrix}and\begin{pmatrix} 1& 0&0\\ 0 &1&0\\0&0&1 \end{pmatrix}$$

viii) Zero Matrix or Null Matrix

A matrix having each of the elements zero is called a zero matrix.For example,

$$\begin{pmatrix} 0 & 0\\ 0 &0 \end{pmatrix}and\begin{pmatrix} 0& 0&0\\ 0 &0&0\\0&0&0 \end{pmatrix}$$

ix) Triangular Matrix

A triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.For example,

$$\begin{pmatrix} 6& 0\\ 8&-1 \end{pmatrix}and\begin{pmatrix} 2& 0&0\\ 3 &-1&0\\7&-3&1 \end{pmatrix}$$

These matrices are said to be lower matrices as all the elements above the main diagonal are zero.

$$\begin{pmatrix} 2& 4\\ 0 &-1 \end{pmatrix}and\begin{pmatrix} 2& -1&3\\ 0 &4&5\\0&0&1 \end{pmatrix}$$

These matrices are said to be lower matrices as all the elements belowthe main diagonal are zero.

x) Submatrix of Matrices

Any matrix obtained by omitting some rows and columns from a given matrix. For example,

\begin{pmatrix}
1 & 0& 3\\
1 &0 & 6
\end{pmatrix}is a submatrix of, \begin{pmatrix}
1& 0&3&7\\ 1&0&6&3\\
2 &1&-1&4\end{pmatrix}

##### Things to remember
• A rectangular array of numbers arranged in rows (horizontal lines) and columns (vertical lines) enclosed between round (or square) brackets is called a matrix.
• The size of the matrix or the order of the matrix is defined as the number of rows followed by the number of columns.
• There are different types of matrices like row matrix,column matrix,square matrix, etc.
• A matrix which has only one row is called a row matrix.
• A matrix which has only one column is called a column matrix.
• A matrix having the same number of row and columns is called a square matrix.
• A matrix for which horizontal and vertical dimensions are not the same is called a rectangular matrix.
• A square matrix which has zeros everywhere other than the main diagonal is called a diagonal matrix.
• A diagonal matrix having all the diagonal elements equal is called a scalar matrix.
• A diagonal matrix having all the diagonal elements equal to 1 is called a unit matrix.

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