Question

Find rank of a unit matrix which is of order $\left( {9 \times 9} \right)$. If $I$ is $\left( {9 \times 9} \right)$unit matrix, then rank $\left( I \right)$=
$
  {\text{A}}{\text{. 0}} \\
  {\text{B}}{\text{. 3}} \\
  {\text{C}}{\text{. 6}} \\
  {\text{D}}{\text{. 9}} \\
 $

Answer 1

Hint:We have given a unit matrix and we should have knowledge that the determinant of a unit matrix of any order is 1. We should also understand that on converting in echelon form we can easily find the rank of a matrix.

Complete step-by-step answer:
Given a unit matrix of order 9 and we know that the determinant of the unit matrix of any order is equal to 1.
Rank of a unit matrix of order n is n. For example : let us take an identity matrix or unit matrix of order $3 \times 3$.we can see that it is an echelon form or triangular form. Now we know that the number of non zero rows of the reduced echelon form is the rank of the matrix.
Echelon form: in linear algebra a matrix is in echelon form if each row containing a non zero number has the number ‘1’ appearing in the rows first non zero column. Such an entry will be referred to as leading one .any row which contains all zero is below the rows which contain a non zero entry.
So rank of matrix is it’s order.
Hence here rank is 9 so option D is the correct option.

Note: Whenever we get this type of question the key concept of solving is we should have knowledge of how to find the rank of the matrix. How to convert in an echelon form of matrix to find it’s rank easily. And many more methods of finding rank of matrix depending on type of questions.