A square matrix A is said to be idempotent if A2 = A. An idempotent matrix is non singular iffa)all eigenvalues are realb)all eigenvalues are non-singularc)all eigenvalues are either 1 or 0.d)all eigenvalues are 1Correct answer is option 'D'. Can you explain this answer?

Question

Answer

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Answer

Explanation:

Idempotent Matrix:
An idempotent matrix A is one that when multiplied by itself gives the original matrix, i.e., A^2 = A.

Non-Singular Idempotent Matrix:
An idempotent matrix is non-singular if all its eigenvalues are equal to 1.

Explanation:
- For an idempotent matrix A, A^2 = A.
- Let λ be an eigenvalue of A with corresponding eigenvector x.
- By definition of eigenvalues and eigenvectors, we have Ax = λx.
- Multiplying both sides by A, we get A(Ax) = A(λx) => A^2x = λAx.
- Since A^2 = A (idempotent matrix), we have Ax = λAx.
- This simplifies to (λ - 1)Ax = 0.
- Since x is an eigenvector and cannot be zero, we have (λ - 1) = 0 => λ = 1.
- Therefore, all eigenvalues of an idempotent matrix must be 1 for it to be non-singular.

Conclusion:
In conclusion, an idempotent matrix is non-singular if all its eigenvalues are equal to 1. This property holds true for any idempotent matrix, and it is a key characteristic of non-singular idempotent matrices.

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