Misc 2 - Chapter 3 Class 12 Matrices
Last updated at April 16, 2024 by Teachoo
Proof using property of transpose
Chapter 3 Class 12 Matrices
Concept wise
- Formation and order of matrix
- Types of matrices
- Equal matrices
- Addition/ subtraction of matrices
- Statement questions - Addition/Subtraction of matrices
- Multiplication of matrices
- Statement questions - Multiplication of matrices
- Solving Equation
- Finding unknown - Element
- Finding unknown - Matrice
- Transpose of a matrix
- Symmetric and skew symmetric matrices
-
Proof using property of transpose
- Inverse of matrix using elementary transformation
- Proof using mathematical induction
Transcript
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.
