Matrix Types: Matrix - Mathematics (Grade 11)

Row Matrix

A row matrix is a matrix that consists of only one row.

The order of a row matrix is $1 \times n$ , where $n$ is the number of columns.

Its general form is:

A_{1 \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \end{bmatrix}

Example:

P = \begin{bmatrix} 3 & -1 & 0 & 7 \end{bmatrix}

Matrix $P$ is a row matrix of order $1 \times 4$ .

Column Matrix

A column matrix is a matrix that consists of only one column.

The order of a column matrix is $m \times 1$ , where $m$ is the number of rows.

Its general form is:

B_{m \times 1} = \begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix}

Example:

Q = \begin{bmatrix} 2 \\ 5 \\ -4 \end{bmatrix}

Matrix $Q$ is a column matrix of order $3 \times 1$ .

Square Matrix

A square matrix is a matrix that has the same number of rows and columns.

If the number of rows = number of columns = $n$ , then the matrix is of order $n \times n$ .

Its general form is:

A_{n \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{bmatrix}

In a square matrix, there are:

Main Diagonal (or Principal Diagonal):

The elements $a_{11}, a_{22}, \dots, a_{nn}$ (i.e., $a_{ij}$ where $i=j$ ).
Anti-diagonal (or Counter-diagonal):

The elements $a_{1n}, a_{2,n-1}, \dots, a_{n1}$ (i.e., $a_{ij}$ where $i+j=n+1$ ).

Example:

M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

Matrix $M$ is a square matrix of order $3 \times 3$ . Its main diagonal elements are $1, 5, 9$ . Its anti-diagonal elements are $3, 5, 7$ .

Rectangular Matrix

A rectangular matrix is a matrix where the number of rows and columns are not equal ( $m \neq n$ ).

General example:

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

Matrix $C$ above has 2 rows and 3 columns, so its order is $2 \times 3$ . Since the number of rows is not equal to the number of columns ( $2 \neq 3$ ), matrix $C$ is a rectangular matrix.

Rectangular matrices can be further distinguished into horizontal matrices and vertical matrices.

Horizontal Matrix

A horizontal matrix is a rectangular matrix with more columns than rows ( $n > m$ ).

Example:

D = \begin{bmatrix} 1 & 0 & 4 \\ -2 & 3 & 5 \end{bmatrix}

Matrix $D$ is a horizontal matrix of order $2 \times 3$ .

Vertical Matrix

A vertical matrix is a rectangular matrix with more rows than columns ( $m > n$ ).

Example:

T = \begin{bmatrix} 7 & 1 \\ 0 & -3 \\ 4 & 2 \end{bmatrix}

Matrix $T$ is a vertical matrix of order $3 \times 2$ .

Triangular Matrix

A triangular matrix is a square matrix where the elements below or above the main diagonal are zero.

Upper Triangular Matrix

An upper triangular matrix is a square matrix where all elements below the main diagonal are zero.

This means $a_{ij} = 0$ for every $i > j$ .

Example:

U = \begin{bmatrix} 5 & 2 & -1 \\ 0 & 3 & 7 \\ 0 & 0 & 1 \end{bmatrix}

Lower Triangular Matrix

A lower triangular matrix is a square matrix where all elements above the main diagonal are zero.

This means $a_{ij} = 0$ for every $i < j$ .

Example:

L = \begin{bmatrix} 2 & 0 & 0 \\ 4 & 6 & 0 \\ -3 & 1 & 9 \end{bmatrix}

Diagonal Matrix

A diagonal matrix is a square matrix where all elements outside the main diagonal are zero.

This means $a_{ij} = 0$ for every $i \neq j$ . Elements on the main diagonal can be zero or non-zero.

Example:

X = \begin{bmatrix} 7 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Matrix $X$ is a diagonal matrix of order $3 \times 3$ .

Identity Matrix

An identity matrix (denoted by $I$ or $I_n$ ) is a diagonal matrix where all elements on the main diagonal are 1.

Example:

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}

The identity matrix acts as the neutral element in matrix multiplication.

Zero Matrix

A zero matrix (denoted by $O$ or $O_{m \times n}$ ) is a matrix where all elements are zero.

Example:

O_{2 \times 2} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

O_{2 \times 3} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose ( $A^T = A$ ).

This means the element $a_{ij} = a_{ji}$ for all $i$ and $j$ . Its elements are symmetric with respect to the main diagonal.

Example:

S = \begin{bmatrix} 1 & 7 & -3 \\ 7 & 2 & 0 \\ -3 & 0 & 5 \end{bmatrix}

In matrix $S$ :

$s_{12} = s_{21} = 7$
$s_{13} = s_{31} = -3$
$s_{23} = s_{32} = 0$

Row Matrix

A row matrix is a matrix that consists of only one row.

The order of a row matrix is $1 \times n$ , where $n$ is the number of columns.

Its general form is:

A_{1 \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \end{bmatrix}

Example:

P = \begin{bmatrix} 3 & -1 & 0 & 7 \end{bmatrix}

Matrix $P$ is a row matrix of order $1 \times 4$ .

Column Matrix

A column matrix is a matrix that consists of only one column.

The order of a column matrix is $m \times 1$ , where $m$ is the number of rows.

Its general form is:

B_{m \times 1} = \begin{bmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{bmatrix}

Example:

Q = \begin{bmatrix} 2 \\ 5 \\ -4 \end{bmatrix}

Matrix $Q$ is a column matrix of order $3 \times 1$ .

Square Matrix

A square matrix is a matrix that has the same number of rows and columns.

If the number of rows = number of columns = $n$ , then the matrix is of order $n \times n$ .

Its general form is:

A_{n \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{bmatrix}

In a square matrix, there are:

Main Diagonal (or Principal Diagonal):

The elements $a_{11}, a_{22}, \dots, a_{nn}$ (i.e., $a_{ij}$ where $i=j$ ).
Anti-diagonal (or Counter-diagonal):

The elements $a_{1n}, a_{2,n-1}, \dots, a_{n1}$ (i.e., $a_{ij}$ where $i+j=n+1$ ).

Example:

M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

Matrix $M$ is a square matrix of order $3 \times 3$ . Its main diagonal elements are $1, 5, 9$ . Its anti-diagonal elements are $3, 5, 7$ .

Rectangular Matrix

A rectangular matrix is a matrix where the number of rows and columns are not equal ( $m \neq n$ ).

General example:

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

Matrix $C$ above has 2 rows and 3 columns, so its order is $2 \times 3$ . Since the number of rows is not equal to the number of columns ( $2 \neq 3$ ), matrix $C$ is a rectangular matrix.

Rectangular matrices can be further distinguished into horizontal matrices and vertical matrices.

Horizontal Matrix

A horizontal matrix is a rectangular matrix with more columns than rows ( $n > m$ ).

Example:

D = \begin{bmatrix} 1 & 0 & 4 \\ -2 & 3 & 5 \end{bmatrix}

Matrix $D$ is a horizontal matrix of order $2 \times 3$ .

Vertical Matrix

A vertical matrix is a rectangular matrix with more rows than columns ( $m > n$ ).

Example:

T = \begin{bmatrix} 7 & 1 \\ 0 & -3 \\ 4 & 2 \end{bmatrix}

Matrix $T$ is a vertical matrix of order $3 \times 2$ .

Triangular Matrix

A triangular matrix is a square matrix where the elements below or above the main diagonal are zero.

Upper Triangular Matrix

An upper triangular matrix is a square matrix where all elements below the main diagonal are zero.

This means $a_{ij} = 0$ for every $i > j$ .

Example:

U = \begin{bmatrix} 5 & 2 & -1 \\ 0 & 3 & 7 \\ 0 & 0 & 1 \end{bmatrix}

Lower Triangular Matrix

A lower triangular matrix is a square matrix where all elements above the main diagonal are zero.

This means $a_{ij} = 0$ for every $i < j$ .

Example:

L = \begin{bmatrix} 2 & 0 & 0 \\ 4 & 6 & 0 \\ -3 & 1 & 9 \end{bmatrix}

Diagonal Matrix

A diagonal matrix is a square matrix where all elements outside the main diagonal are zero.

This means $a_{ij} = 0$ for every $i \neq j$ . Elements on the main diagonal can be zero or non-zero.

Example:

X = \begin{bmatrix} 7 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Matrix $X$ is a diagonal matrix of order $3 \times 3$ .

Identity Matrix

An identity matrix (denoted by $I$ or $I_n$ ) is a diagonal matrix where all elements on the main diagonal are 1.

Example:

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}

The identity matrix acts as the neutral element in matrix multiplication.

Zero Matrix

A zero matrix (denoted by $O$ or $O_{m \times n}$ ) is a matrix where all elements are zero.

Example:

O_{2 \times 2} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

O_{2 \times 3} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose ( $A^T = A$ ).

This means the element $a_{ij} = a_{ji}$ for all $i$ and $j$ . Its elements are symmetric with respect to the main diagonal.

Example:

S = \begin{bmatrix} 1 & 7 & -3 \\ 7 & 2 & 0 \\ -3 & 0 & 5 \end{bmatrix}

In matrix $S$ :

$s_{12} = s_{21} = 7$
$s_{13} = s_{31} = -3$
$s_{23} = s_{32} = 0$

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