Question

How do you classify matrices?

Answer 1

Matrix :

A matrix is a rectangular arrangement of numbers into rows and columns.

Example: ${\left[\begin{array}{cc}2& 3\\ 5& -6\end{array}\right]}_{2\times 2}$

Classification of matrices:

The Matrices are classified on the basis of the number of rows and columns, and the specific elements within them.

Row Matrix: A matrix that has exactly one row is called a row matrix.

Example: $\left[\begin{array}{ccc}-1& 0& 2\end{array}\right],\left[\begin{array}{cc}5& 4\end{array}\right],\left[\begin{array}{c}2\end{array}\right]$etc. all have a row.

Column Matrix: A matrix that has exactly one column is called a column matrix.

Example : $\left[\begin{array}{c}4\\ 3\\ 2\\ 1\end{array}\right]$,$\left[\begin{array}{c}5\\ 4\\ -8\end{array}\right]$ etc all have 1 column.

Rectangular Matrix: A matrix of the order m × n in which m ≠ n is called a rectangular matrix.

Example: $\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right],\left[\begin{array}{ccc}5& 7& 6\\ 4& 5& 0\end{array}\right]$ etc. all have different row and column

Zero Matrix or Null Matrix: A matrix is called a zero matrix if all its elements are zeros.

Example: $\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$etc.

Square Matrix: A matrix of the order m × n in which m = n is called a square matrix of order m (or n).

Example: $\left[\begin{array}{cc}-1& 2\\ 0& 4\end{array}\right],\left[\begin{array}{ccc}0& 8& 9\\ 4& 0& 4\\ 5& 3& 1\end{array}\right]$etc. having the same number of rows and columns.

Unit Matrix or Identity Matrix: A square matrix in which all elements in the leading diagonal are 1 and the other elements are zeros, is called a unit matrix.

Example: $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$etc. as all elements in diagonal is $1$and other are $0$.

Diagonal matrix: A square matrix in which every element except the principal diagonal elements is zero i

Example: $\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right],\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 8\end{array}\right]$etc.

Triangular Matrix: A square matrix in which elements below and/or above the diagonal are all zeros. There are two types of triangular matrices

1. Lower triangular matrix: A square matrix whose all elements above the main diagonal are zero

Example: $\left[\begin{array}{ccc}-8& 0& 0\\ 2& 5& 0\\ 1& 3& 2\end{array}\right],\left[\begin{array}{cccc}4& 0& 0& 0\\ 12& 5& 0& 0\\ 0& 4& 6& 0\\ 9& 5& 0& 7\end{array}\right]$etc.

2. Upper triangular matrix: A square matrix whose all elements below the main diagonal are zero

Example: $\left[\begin{array}{ccc}-8& 5& 9\\ 0& 5& 1\\ 0& 0& 2\end{array}\right],\left[\begin{array}{cccc}4& 5& 2& 0\\ 0& 5& 1& 0\\ 0& 0& 6& 9\\ 0& 0& 0& 7\end{array}\right]$etc. :

Symmetric matrix: A square matrix that is equal to its transpose.

Example: If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$then $A^t=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$So,$A=A^t$

Anti symmetric matrix: A square matrix whose transpose is equal to its negative also known as a skew symmetric matrix.

Example: If $A=\left[\begin{array}{cc}0& 5\\ -5& 0\end{array}\right],A^t=\left[\begin{array}{cc}0& -5\\ 5& 0\end{array}\right]$So, $A=-A^t$

Hence, matrices are classified on the basis of the number of rows and columns, and the specific elements within them.