Idempotent Matrix
Question 1: What is meant by an idempotent matrix?
Answer:
An idempotent matrix is defined as a square matrix that remains unchanged when multiplied by itself.
Question 2: How to prove that a matrix is idempotent?
Answer:
Any square matrix “P” is said to be an idempotent matrix if and only if P2 = P. So, to prove that a matrix is idempotent, then the matrix must satisfy the above condition.
Question 3: Does the inverse of an idempotent matrix exist?
Answer:
We know that the inverse of a square “A” (A-1) = Adj A/ |A|
If the given idempotent matrix is singular, then its inverse does not exist as its determinant is zero.
Question 4: What is the relationship between an idempotent matrix and an involuntary matrix?
Answer:
The following is the relationship between idempotent and involuntary matrices: A square matrix “A” is said to be an idempotent matrix if and only if P = 2A − I is an involuntary matrix.
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.