Misc 2 - Chapter 3 Class 12 Matrices

Last updated at June 8, 2023 by

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Misc 2 Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric. We need to prove B’AB is symmetric if A is symmetric and B’AB is skew symmetric if A is skew symmetric Proving B’AB is symmetric if A is symmetric Let A be a symmetric matrix, then A’ = A Taking (B’AB)’ Let AB = P = (B’P)’ = P’ (B’)’ = P’ B Putting P = AB = (AB)’ (B) = B’A’ (B) = B’AB ∴ (B’AB)’ = B’AB Thus, B’AB is a symmetric matrix Proving B’AB is skew-symmetric if A is skew-symmetric Let A be a skew-symmetric matrix, then A’ = – A Taking (B’AB)’ Let AB = P = (B’P)’ = P’ (B’)’ = P’ B Putting P = AB = (AB)’ (B) = B’A’ (B) = B’(–A)B = – B’AB ∴ (B’AB)’ = – B’AB Thus, B’AB is a skew symmetric matrix Hence, matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.