Q. 53.6( 32 Votes )

Show that the mat

Answer :

Case 1: When A is a symmetric matrix i.e.

A = A’……. (1)


where A’ is the transpose of A


To prove: B’AB is also a symmetric matrix.


Calculating the transpose of B’AB


(B’AB)’= B’A’(B’)’ (By property of transpose i.e. (AB)’ = B’A’)


= B’A’B (By property of transpose i.e. (A’)’ = A)


= B’AB (from (1))


It satisfies the condition of symmetric matrix as matrix B’AB is equal to its transpose.


Hence B’AB is a symmetric matrix when A is symmetric.


Case 2: When A is a skew symmetric matrix i.e.


A = -A’……. (2)


where A’ is the transpose of A.


To prove: B’AB is also a skew symmetric matrix.


Calculating the transpose of B’AB


(B’AB)’= B’A’(B’)’ (By property of transpose i.e. (AB)’ = B’A’)


= B’A’B (By property of transpose i.e. (A’)’ = A)


= B’(-A)B (from (2))


= - (B’AB)


It satisfies the condition of skew symmetric matrix as matrix (B’AB) is equal to its transpose.


Hence (B’AB) is a skew symmetric matrix when A is skew symmetric.


Both results are proved.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

<span lang="EN-USMathematics - Exemplar

<span lang="EN-USMathematics - Exemplar

Fill in the blankMathematics - Exemplar

Fill in the blankMathematics - Exemplar

Fill in the blankMathematics - Exemplar

If A, B are squarMathematics - Exemplar

Fill in the blankMathematics - Exemplar

Fill in the blankMathematics - Exemplar

Express the folloMathematics - Board Papers

Fill in the blankMathematics - Exemplar