Question

What are the types of Matrices?

Answer 1

Types of Matrices:

Matrix is an arrangement of elements in rows and columns.

The types of matrices are given below.

• Row matrix:

The type matric in which there is only one row and $n$number of columns. The general form of the matrix is ${\left[a_{ij}\right]}_{1\times n}$.

The order of the matrix is $1\times n$.

• Column matrix:

The type matric in which there is only one column and $n$number of rows. The general form of the matrix is ${\left[a_{ij}\right]}_{m\times 1}$.

The order of the matrix is $m\times 1$.

• Square matrix:

The type matrix in which the number of rows and columns are equal. The general form of the matrix is ${\left[a_{ij}\right]}_{m\times m}$.

The order of the matrix is $m.$

• Rectangular matrix:

The type matrix in which there is an unequal number of rows and columns. The general form of the matrix is ${\left[a_{ij}\right]}_{m\times n}$.

The order of the matrix is $m\times n$.

• Diagonal Matrix:

A square matrix is said to be a diagonal matrix if the non-diagonal elements are zero.

For example, matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]$ is a diagonal matrix.

• Scalar Matrix:

A diagonal matrix is said to be a scalar matrix if the diagonal elements are the same.

For example, matrix $A=\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 5\end{array}\right]$ is a scalar matrix.

• Null Matrix:

A matrix is said to be a zero or null matrix if all the elements in the matrix are zero.

For example, matrix $O=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$ is a null matrix.

• Identity Matrix:

A square matrix is said to be an identity or unit matrix if the non-diagonal elements are zero and the diagonal elements are unity.

For example, matrix $I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ is an identity matrix.

• Matrix of ones:

A matrix of ones is a matrix in which all the elements are equal to unity.

For example, matrix $C=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ is a matrix of ones.

• Singular matrix:

A singular matrix is a matrix and the determinant value of the matrix is equal to zero.

• Non-singular matrix:

A singular matrix is a matrix in which the determinant value is a non-zero number.

• Upper triangular matrix:

An upper triangular matrix is a square matrix in which the elements below the diagonals are zero.

For example, matrix $U=\left[\begin{array}{ccc}1& -6& -5\\ 0& 7& 8\\ 0& 0& -9\end{array}\right]$

• Lower triangular matrix:

A lower triangular matrix is a square matrix in which the elements above the diagonals are zero.

For example, matrix $L=\left[\begin{array}{ccc}1& 0& 0\\ 2& 7& 0\\ 3& 9& 5\end{array}\right]$

• Symmetric Matrix:

A square matrix is said to be symmetric if and only if it is equal to the transpose.

For example, $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 5\end{array}\right]$ and $A^T=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 5\end{array}\right]$

Here $A=A^T$

• Skew Symmetric Matrix:

A square matrix $S$is said to be skew-symmetric if and only if $S^T=-S$.

For example, $S=\left[\begin{array}{cc}0& 4\\ -4& 0\end{array}\right]$ and $S^T=\left[\begin{array}{cc}0& -4\\ 4& 0\end{array}\right]$

Here $S^T=-S$.

• Orthogonal Matrix:

A square matrix is an orthogonal matrix of the product of the matrix and its transpose is equal to the identity matrix i.e, $A\times A^T=I$.

• Boolean Matrix:

A matrix in which all the elements are either zero or unity.

For example, $B=\left[\begin{array}{ccc}1& 0& 1\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ is a Boolean matrix.

• Hermitian Matrix:

It is a square matrix that is equal to its own conjugate transpose matrix.$A=\overline{A^T}$.

The elements of the Hermitian matrix are complex numbers.

• Skew Hermitian Matrix:

A square matrix is a skew Hermitian matrix if and only if its conjugate transpose is equal to its negative.$-A=\overline{A^T}$.

A skew Hermitian matrix is also called the anti-Hermitian matrix.