Let A be a 4 x 4 matrix with real entries such that - 1, 1, 2, - 2 are its eigen values.If B = A4 - 5A2 + 5I, where I denotes the 4 x 4 identity matrix, then which of thefollowing statements are correct?a)det(A + B) = 0b)det (B) = 1c)trace of A - B is 0d)trace of A + B is 4Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Given information:
- A is a 4x4 matrix with real entries.
- The eigenvalues of A are -1, 1, 2, -2.
- B = A^4 - 5A^2 + 5I, where I is the 4x4 identity matrix.

To find the correct statements, let's analyze each option:

a) det(A B) = 0
To determine det(A B), we need to compute the determinant of the matrix [A B]. Since B is defined in terms of A, let's substitute the expression for B into the matrix:

[A B] = [A (A^4 - 5A^2 + 5I)]
[ 0 5I ]

Using the block determinant formula, we have:

det([A B]) = det(A) det(A^4 - 5A^2 + 5I) - det(A (A^4 - 5A^2 + 5I) 0)
= det(A) det(A^4 - 5A^2 + 5I)

Since the eigenvalues of A are -1, 1, 2, and -2, we know that det(A) = (-1)(1)(2)(-2) = 4.

Substituting this value in, we get:

det([A B]) = 4 det(A^4 - 5A^2 + 5I)

Since the determinant of a scalar multiple of a matrix is equal to the scalar raised to the power of the matrix's dimension, we can simplify further:

det([A B]) = 4 det(A^4 - 5A^2 + 5I) = 4^4 det(A^4 - 5A^2 + 5I)
= 4^4 det(A^4) det(A^4 - 5A^2 + 5I)
= 4^4 (det(A))^4 det(A^4 - 5A^2 + 5I)
= 4^4 (4)^4 det(A^4 - 5A^2 + 5I)

Since det(A^4 - 5A^2 + 5I) is a scalar, we can factor it out:

det([A B]) = (4^4 (4)^4) det(A^4 - 5A^2 + 5I)
= 4^8 det(A^4 - 5A^2 + 5I)

Therefore, det(A B) is a non-zero value, so statement a) is incorrect.

b) det(B) = 1
To find the determinant of B, we substitute the expression for B:

B = A^4 - 5A^2 + 5I

Taking the determinant of both sides:

det(B) = det(A^4 - 5A^2 + 5I)

Since I is the identity matrix, its determinant is 1. Therefore, we have:

det(B) = det(A^4 - 5A^2 + 5I) = det(A^4 - 5A^2 + 5)
= det(A^4) - 5 det(A^2) + det(5I)

Since A is a 4x4 matrix, we know that

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