InterviewSolution
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InterviewSolution
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Show that the matrix `B^TA B`is symmetric or skew-symmetric according as A is symmetric orskew-symmetric. |
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Answer» A matrix `A` is symmetric if `A = A^T`. A matrix `A` is skew-symmetric if `A = -A^T`. (i) When `A` is symmetric matrix, then `A = A^T` Then, `(B^TAB)^T = B^TA^T(B^T)^T = B^TAB`...[As `A^T = A and (B^T)^T = B`] `:. (B^TAB)^T = B^TAB` `:. B^TAB` is a symmetric matrix when `A` is symmetric (ii)When `A` is skew-symmetric matrix, then `A = -A^T` Then, `(B^TAB)^T = B^TA^T(B^T)^T = B^T(-A)B`...[As `A^T =- A and (B^T)^T = B`] `:. (B^TAB)^T =- B^TAB` `:. B^TAB` is a skew-symmetric matrix when `A` is skew-symmetric |
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