Let I be the identity matrix of order 3×3 and let A=λ4725−1362 with ∣A∣=−1. Let B be the inverse of the matrix adj(A)adj(A2). Then ∣λB+I∣ is equal to:
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Step-by-Step Solution
Step 1: Simplify the expression for B
We are given that B is the inverse of the matrix adj(A)adj(A2). We will use properties of adjoints and inverses to simplify this expression.
Step 2: Apply properties of adjoints and inverses
We use the property that adj(An)=(adj(A))n. Also, recall that adj(A)=∣A∣A−1 and the inverse of a product of matrices is the product of their inverses in reverse order.
Step 3: Substitute and simplify B
First, we combine the adjoint terms: adj(A)(adj(A))2=(adj(A))3. Then, B=((adj(A))3)−1=(adj(A))−3. Since ∣A∣=−1, we have adj(A)=∣A∣A−1=−A−1. Substituting this, B=(−A−1)−3=(−1)−3(A−1)−3=−A3.
Step 4: Calculate ∣λB+I∣
Now we substitute B=−A3 into the expression ∣λB+I∣.
Step 5: Use the property ∣kA∣=kn∣A∣
We know that ∣A∣=−1. For a 3×3 matrix M and a scalar k, the determinant of kM is k3∣M∣.
Step 6: Final calculation
The expression ∣−λA3+I∣ cannot be simplified further without knowing the value of λ or the matrix A. The problem statement implies that the answer should be a specific numerical value, which suggests there might be a misunderstanding or missing information in the problem statement as presented. However, if we assume that the question implies a property that makes this expression simplify to a constant, it's not immediately obvious. Given the options usually provided in such problems, it's possible that the question expects a specific property of determinants that is not directly applicable here, or there's a typo in the question. Without further information or context, we cannot simplify ∣−λA3+I∣ to a numerical value.