QQ.1. Answer the following questions.
- (a)) If A, B are symmetric matrices of the same order, then what will be the type of matrix (AB – BA)? Give reasons in support of your answer. (50 words)
- (b)) If the matrix A is both symmetric and skew-symmetric, then determine A. (50 words)
- (c)) If A is a square matrix, such that A² =A, then find (I + A)³ - 7A. (50 words)
- (d)) If A = [[1 0 0], [0 1 -2], [1 0 4]] ; I = [[1 0 0], [0 1 0], [0 0 1]], A⁻¹ = 1/6 (A² + aA + bI). Find a & b. (100 words)
- Skew-symmetric matrix: Mᵀ = -M.
- Symmetric matrix: Mᵀ = M.
- Zero matrix is the only matrix that is both symmetric and skew-symmetric.
- Idempotent matrix property: If A² = A, then (I+A)³ = I+7A.
Answer: This question assesses fundamental concepts and applications in Linear Algebra, specifically properties of matrices and the Cayley-Hamilton Theorem. It covers identifying matrix types based on transpose properties, understanding idempotent matrices, and deriving a matrix inverse using its characteristic equation. **(a) Type of matrix (AB – BA)** If A and B are symmetric matrices, then Aᵀ = A and Bᵀ = B. Let M = (AB - BA). To determine its type, we compute its transpose: Mᵀ = (AB - BA)ᵀ. Using t...