Types of Matrices

A matrix is a rectangular array of numbers, arranged in rows and columns. Here are two of such matrices:

\[\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix},\quad \begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}.\]

In this wiki, we are only going to discuss different types of matrices.

Here we will discuss square matrix, horizontal matrix, vertical matrix, row matrix, column matrix, null matrix, diagonal matrix, and scalar matrix.

Square Matrix:

Take a rectangular matrix \(A = (a_{ij})_{m \times n}\) of order \(m \times n\). If \(m = n,\) then matrix \(A\) is said to be a square matrix. In other words, if in a matrix the number of rows and columns are equal, then the matrix is said to be a square matrix. Given below are a few examples of square matriices:

\[\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix},\quad \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{pmatrix},\quad \begin{pmatrix} x & y \end{pmatrix}.\]

Horizontal Matrix:

Take a rectangular matrix \(A = (a_{ij})_{m \times n}\) of order \(m \times n\). If \(m < n,\) then matrix \(A\) is said to be a horizontal matrix. In other words, if in a matrix the number of rows is less in number than the number of columns, then that matrix is said to be a horizontal matrix. Given below are a few examples of horizontal matriices:

\[\begin{pmatrix} a & b \\ x & y \\ c & d \\ \end{pmatrix},\quad \begin{pmatrix} 2 \\ 4 \\ 12 \\ \end{pmatrix},\quad \begin{pmatrix} 1 & 0 \\ 3 & 9 \\ 7 & 2 \\ 5 & 8 \\ \end{pmatrix}.\]

Vertical Matrix:

Take a rectangular matrix \(A = (a_{ij})_{m \times n}\) of order \(m \times n\). If \(m > n,\) then matrix \(A\) is said to be a vertical matrix. In other words, if in a matrix the number of rows is more in number than the number of columns, then that matrix is said to be a vertical matrix. Given below are a few examples of horizontal matriices:

\[\begin{pmatrix} a & b & c \\ x & y & d \\ \end{pmatrix},\quad \begin{pmatrix} 2 & 4 & 12 \end{pmatrix},\quad \begin{pmatrix} 1 & 0 & 7 & 5 \\ 3 & 9 & 2 & 8 \\ 7 & 2 & 3 & 10 \\ \end{pmatrix}.\]

Row Matrix:

Take a rectangular matrix \(A = (a_{ij})_{m \times n}\) of order \(m \times n\). If \(m = 1,\) then matrix \(A\) is said to be a row matrix. In other words, if in a matrix there is only one row present, then that matrix is said to be a row matrix. Given below are a few examples of row matriices:

\[\begin{pmatrix} 1 \\ 7 \\ 4 \\ \end{pmatrix},\quad \begin{pmatrix} p \\ q \\ r \\ \end{pmatrix},\quad \begin{pmatrix} 4 \\ 3 \\ \end{pmatrix}.\]

Column Matrix:

Take a rectangular matrix \(A = (a_{ij})_{m \times n}\) of order \(m \times n\). If \(n = 1,\) then matrix \(A\) is said to be a column matrix. In other words, if in a matrix there is only one coloum present, then that matrix is aid to be a column matrix. Given below are a few examples of row matrices:

\[\begin{pmatrix} 1 & 7 & 4 \\ \end{pmatrix},\quad \begin{pmatrix} p & q & r \\ \end{pmatrix},\quad \begin{pmatrix} 4 & 3 \\ \end{pmatrix}.\]

Null Matrix:

In any matrix if each and every element is \(0\), it is called a null matrix or zero matrix. Given below are 2 null matrices of order \(2 \times 2\) and \(3 \times 3,\) respectively:

\[\begin{pmatrix} 0 & 0 \\ 0 & 0 \\ \end{pmatrix},\quad \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}.\]

Diagonal Matrix:

A square matrix in which all non-diagonal elements are \(0\) is called a diagonal matrix. The diagonal elements also may be \(0\) or may not be \(0,\) but at least there should be one non-zero element. Given below are 2 such matrices:

\[\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix},\quad \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}.\]

Scalar Matrix:

A diagonal matrix in which all principle diagonal elements are equal is called a scalar matrix. Given below are a few examples of scalar matrices:

\[\begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix},\quad \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{pmatrix}.\]

A diagonal matrix in which all principal diagonal values are equal to 1 is called as an identity matrix or a unit matrix. It is denoted by the letter \(I\). Given below is an Identity matrix of order \(3 \times 3\):

\[I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix},\]

where the subscript \(3\) in \(I_3\) tells us that it is an identity matrix of order \(3 \times 3\). Similarly,

\[I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}.\]

Properties:

You can compare an identity matrix \(I\) in matrices and normal \(1\) in mathematics. They both play a similar role as follows:

1. An identity matrix raised to a power of any positive integer will not change but remain as it is: \[\begin{align} I^2 &= I \\ I^3 &= I \\ I^4 &= I \\ &\vdots \\ I^n &= I. \end{align}\]
2. When any matrix is multiplied with an identity matrix, the values in the matrix will not change: \(I \times A = A.\)

Here we will discuss matrices which are idempotent, nilpotent, involuntary, periodic, and singular.

Idempotent Matrix:

Consider a matrix \(A = (a_{ij})_{m \times n}\) of order \(m \times n\). If \(A^2 = A\), then matrix \(A\) is said to be Idempotent.

The best example for an idempotent matrix is the unit matrix itself:

\[I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix},\quad I^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}.\]

Verify whether \(A = \begin{pmatrix} 2 & -2 & -4 \\ 1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{pmatrix}\) is idempotent or not.

ANSWER

To check whether \(A\) is idempotent or not, we should first find \(A^2:\)

\[\begin{align} A^2 &= \begin{pmatrix} 2 & -2 & -4 \\ 1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{pmatrix} \begin{pmatrix} 2 & -2 & -4 \\ 1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{pmatrix} \\\\ &= \begin{pmatrix} 4 - 2 + 4 & -4 - 6 - 8 & 8 - 8 + 12 \\ 2 + 3 + 4 & - 2 + 9 - 8 & 4 + 12 + 12 \\ 2 - 2 - 3 & -2 - 6 + 6 & 4 - 8 - 9 \\ \end{pmatrix} \\\\ &= \begin{pmatrix} 6 & -18 & 12 \\ 9 & -1 & 28 \\ -3 & -2 & -3 \\ \end{pmatrix}. \end{align}\]

Clearly, \(A^2 \neq A\). Therefore, \(A\) is not idempotent. \(_\square\)

Suppose that matrices \(A\) and \(B\) are both idempotent. Also, \(A\) and \(B\) are both commutative. Verify whether matrix \(AB\) is idempotent or not.

ANSWER

Given below are the conditions mentioned in the question:

\[\begin{align} A^2 &= I \\ B^2 &= I \\ AB &= BA. \end{align}\]

To check whether \(AB\) is idempotent or not, we should first find \((AB)^2:\)

\[\begin{align} (AB)^2 &= (A{\color{blue}B})({\color{blue}A}B) \\ &= A({\color{blue}BA})B \\ &= A({\color{blue}AB})B \qquad \qquad (\text{as AB = BA}) \\ &= (A{\color{blue}A})({\color{blue}B}B)\\ &= \underbrace{A^2}_{\text{equal to A}} \cdot \underbrace{B^2}_{\text{equalt to B}} \\ &= AB. \end{align}\]

Clearly, \((AB)^2 = AB.\) Therefore, \(AB\) is an idempotent matrix. \(_\square\)

If \(A\) is an idempotent matrix, then verify whether matrix \(I - A\) is Idempotent or not, where \(I\) is a unit matrix of the same order as \(A\).

ANSWER

We are given that \(A^2 = A\). As usual, to check whether \((I - A)\) is idempotent or not, we should first find \((I - A)^2:\)

\[\begin{align} (I - A)^2 &= (I - A)(I - A) \\ &= I^2 - IA - AI + A^2 \\ \require{cancel} &= I - A - \cancel{A} + \cancel{A} \qquad \qquad \big(\text{as } A^2 = A , I^2 = I\big) \\ &= I - A. \end{align}\]

Clearly, \((I - A)^2 = I - A\). Therefore, \((I - A)\) is idempotent matrix. \(_\square\)