How to calculate the rank of a matrix

Calculate the rank of the following matrices.
$(a)$

Example to calculate the rank of a matrices

$(b)$

Answer

A good method for calculating the rank of a matrix is to transform the matrix to the reduced row echelon form by the elementary row operation, and count the number of the leading entry (See appendix). In the following, we will answer according to this method.

$(a)$

Let $A$ be an matrix defined as

. We transform $A$ to the reduced row echelon form as follows.

As a result of this transformation, since the matrix has two leading entries, we see that the rank of $A$ is $2$.

$(b)$

Let $B$ be an matrix defined as

. We transform $B$ to the reduced row echelon form as follows.

As a result of this transformation, since the matrix has three leading entries, we see that the rank of $B$ is $3$.

Appendix: Leading entries

The leading entry of a matrix is the first non-zero element in each row.

In this example, the elements surrounded by a square are the leading entries. Therefore, there are two leading entries.