Solved Examples on Idempotent Matrix

Example 1: Verify whether the matrix given below is idempotent or not.

Solution:

To prove that the given matrix is idempotent, we have to prove that P² = P.

Hence, verified.

So, the given matrix P is an idempotent matrix.

Example 2: Verify whether the matrix given below is idempotent or not.

Solution:

To prove that the given matrix is idempotent, we have to prove that B² = B.

Hence, verified.

So, the given matrix B is an idempotent matrix.

Example 3: Give an example of an idempotent matrix of order 2 × 2.

Solution:

We know that a matrix  is said to be an idempotent matrix, if bc = a − a² and d = 1 − a.

Let us consider that a = 5

We have, d = 1 − a

d = 1 − 5 = −4

bc = a − a²

bc = 5 − 25 = −20

Now, let b = 4 and c = −5

So, the matrix is A = 

Example 4: Prove that an identity matrix is an idempotent matrix.

Solution:

To prove that the given matrix is idempotent, we have to prove that I² = I.

Let us consider an identity matrix of order 2 × 2, i.e., 

Hence, proved.

So, an identity matrix is an idempotent matrix.

A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an “m by n” matrix when it has “m” rows and “n” columns and is written as an “m × n” matrix. For example, a matrix of order “5 × 6” has five rows and six columns. We have different types of matrices like rectangular matrices, square matrices, null matrices, triangular matrices, symmetric matrices, etc.