Types of matrices — lesson. Mathematics State Board, Class 10.

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In this section, let us discuss different types of matrices.

1. Row matrix

A matrix is a row matrix when it is made up of just one row and '

$n$ ' number of columns. Row matrices are also called row vectors.

For a matrix

$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & … & a_{1n} \end{bmatrix}$ , the order of the matrix is

$1$

$\times$

$n$ .

Example:

Let us look at a few row matrices.

$A = \begin{bmatrix} \sqrt{2} & \frac{\sqrt 7}{2} & 12 \end{bmatrix}$ , where the order of the matrix is

$1$

$\times$

$3$ .

$A = \begin{bmatrix} 7 & 14 & 21 & 28 & 35 & 42 & 49 \end{bmatrix}$ , where the order of the matrix is

$1$

$\times$

$7$ .

2. Column matrix

A matrix is a column matrix comprised of 'm' rows and just one column. Column matrices are also called column vectors.

For a matrix

$A = \begin{bmatrix} a_{11}\\ a_{21}\\ a_{31}\\ \vdots\\ a_{m1} \end{bmatrix}$ , the order of the matrix is

$m$

$\times$

$1$ .

Example:

Let us look at a few column matrices.

$A = \begin{bmatrix} 2x\\ 3x\\ x \end{bmatrix}$ , where the order of the matrix is

$3$

$\times$

$1$ .

$A = \begin{bmatrix} 4\\ 8\\ 12\\ 16\\ 20 \end{bmatrix}$ , where the order of the matrix is

$5$

$\times$

$1$ .

3. Square matrix

A matrix is a square matrix when the number of rows equals the number of columns. In other words,

$m$

$=$

$n$ .

The order of a square matrix is

$m$ .

Example:

Let us look at a few square matrices.

$A = \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}$ , where the order of the matrix is

$2$

$\times$

$2$ .

$A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}$ , where the order of the matrix is

$3$

$\times$

$3$ .

Leading diagonal of a square matrix:

In a square matrix, the entries

$a_{ij}$ where

$i$

$=$

$j$ form the leading diagonal of a matrix.

Example:

Let us consider the matrix given below.

$A = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix}$

Here, the elements

$a_{11}$ ,

$a_{22}$ and

$a_{33}$ form the elements of the leading diagonal.

4. Diagonal matrix

In a square matrix, when all the entries except the leading diagonal is zero, then it is called a diagonal matrix. In other words,

$a_{ij}$

$=$

$0$ for

$i$

$\neq$

$j$ .

Example:

Let us look at a few diagonal matrices.

$A = \begin{bmatrix} 1 & 0\\ 0 & 4 \end{bmatrix}$

$A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 5 & 0\\ 0 & 0 & 9 \end{bmatrix}$

5. Scalar matrix

In a square matrix, when all the elements of the leading diagonal is the same, then it is a scalar matrix

The general representation of a scalar matrix

$A$

$=$

$(a_{ij})_{m \times n}$ is

$a_{ij} = \begin{cases} 0 & \text{ when } i \neq j \\ k & \text{ when } i = j \end{cases}$ , where

$k$ is a constant.

Example:

Let us look at a few scalar matrices.

$A = \begin{bmatrix} 7 & 0 & 0\\ 0 & 7 & 0\\ 0 & 0 & 7 \end{bmatrix}$

$A = \begin{bmatrix} \sqrt{3} & 0 & 0\\ 0 & \sqrt{3} & 0\\ 0 & 0 & \sqrt{3} \end{bmatrix}$

6. Identity or unit matrix

In a square matrix, when all the leading diagonal elements are

$1$ , then it is an identity matrix or a unit matrix.

The general representation of a unit matrix

$A$

$=$

$(a_{ij})_{m \times n}$ is

$a_{ij} = \begin{cases} 0 & \text{ when } i \neq j \\ 1 & \text{ when } i = j \end{cases}$ .

Example:

Let us look at a few identity matrices.

$\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

$\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$

7. Zero matrix or null matrix

A matrix is a zero matrix or a null matrix when all the matrix elements are zero.

Example:

Let us look at a few null matrices.

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

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

8. Transpose of a matrix

The transpose of a matrix is obtained by interchanging the elements in the rows and columns. The transpose of a matrix is denoted by

$A^T$ , and

$A^T$ is read as '

$A$

$\text{transpose}$ '.

If the order of matrix

$A$ is

$m \times n$ , then the order of

$A^T$ is

$n \times m$ .

Example:

Let us now look at a few examples.

1. If

$A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}$ , then

$A^T = \begin{bmatrix} 1 & 4 & 7\\ 2 & 5 & 8\\ 3 & 6 & 9 \end{bmatrix}$ .

2. If

$B = \begin{bmatrix} 1 & 2 & 3 & 4\\ 5 & 6 & 7 & 8\\ 9 & 10 & 11 & 12 \end{bmatrix}$ , then

$B^T = \begin{bmatrix} 1 & 5 & 9\\ 2 & 6 & 10\\ 3 & 7 & 11\\ 4 & 8 & 12 \end{bmatrix}$ .

9. Triangular matrix

If all the entries above the leading diagonal are zero in a square matrix, then it is a lower triangular matrix. If all the entries below the leading diagonal are zero, then it is an upper triangular matrix.

A matrix

$A$ is a upper triangular matrix if

$a_{ij}$

$=$

$0$ for

$i$

$>$

$j$ . Similarly, matrix

$A$ is a lower triangular matrix if

$a_{ij}$

$=$

$0$ for

$i$

$<$

$j$ .

Example:

Let us look at a few examples.

Upper triangular matrices:

$A = \begin{bmatrix} 1 & 0 & 0\\ 2 & 3 & 0\\ 4 & 5 & 6 \end{bmatrix}$ ,

$B = \begin{bmatrix} 7 & 0 & 0\\ 8 & 9 & 0\\ 10 & 11 & 12 \end{bmatrix}$

Lower triangular matrices:

$C = \begin{bmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 6 \end{bmatrix}$ ,

$D = \begin{bmatrix} 7 & 8 & 9\\ 0 & 10 & 11\\ 0 & 0 & 12 \end{bmatrix}$

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3. Types of matrices

Theory: