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)trace of A - B is 0b)det (B) = 1c)det(A + B) = 0d)trace of A + B is 4Correct answer is option 'C'. Can you explain this answer?

Question

Answer

To solve this question, we need to consider the given matrix A and use its eigenvalues to find the values of B. Let's break down the solution step by step.

Step 1: Finding the matrix A
Given that the eigenvalues of matrix A are -1, 1, 2, and -2, we can write the characteristic equation as follows:
det(A - λI) = 0,
where λ represents the eigenvalues and I is the identity matrix.

Substituting the given eigenvalues into the equation, we have:
det(A + I) = 0,
det(A - I) = 0,
det(A - 2I) = 0,
det(A + 2I) = 0.

Solving these equations, we find the corresponding eigenvectors:
For eigenvalue -1, let's say v1,
For eigenvalue 1, let's say v2,
For eigenvalue 2, let's say v3,
For eigenvalue -2, let's say v4.

Step 2: Finding the matrix B
Now, we need to calculate B = A^4 - 5A^2 + 5I.

To simplify the calculation, let's find A^2 and A^4 first:
A^2 = A * A,
A^4 = A^2 * A^2.

Using the eigenvectors found in Step 1, we can diagonalize matrix A as follows:
A = PDP^(-1),
where P is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues.

Now, we can calculate A^2 and A^4 using the diagonalized form:
A^2 = (PDP^(-1)) * (PDP^(-1)),
A^4 = A^2 * A^2.

Step 3: Calculating B using A^4 and A^2
Now that we have A^4 and A^2, we can substitute them into the expression for B:
B = A^4 - 5A^2 + 5I.

Since B involves A^4 and A^2, we can use the diagonalized form of A to simplify the calculation:
B = (PDP^(-1))^4 - 5(PDP^(-1))^2 + 5I.

Step 4: Simplifying B using diagonalized form
Expanding the expression for B using the diagonalized form of A, we have:
B = (PD^4P^(-1)) - 5(PDP^(-1))(PDP^(-1)) + 5I.

Simplifying further, we get:
B = P(D^4 - 5D^2 + 5I)P^(-1).

Step 5: Calculating the determinant of B
To determine whether det(B) = 0, we need to calculate the determinant of B:
det(B) = det(P(D^4 - 5D^2 + 5I)P^(-1)).

Using the property that det(AB) = det(A)det(B), we can simplify the expression:
det(B) = det(P)det(D^4 - 5D^2 + 5I)det(P^(-1)).

Since P is a matrix of eigenvectors, its determinant is not equal to zero, so we can ignore it

Answer

Here option b,c,d are right

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