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Chapter 5: Q5E (page 263)

TRUE OR FALSE?If A and Bare symmetric n脳nmatrices, then A+B must be symmetric as well.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of a symmetric matrix.

Suppose A is an n脳nmatrix.

Then the matrix is said to be symmetric matrix, ifAT=A.

By the properties of transpose,(A+B)T=AT+BT.

02

Check whether the given statement is a true or false.

The given statement is if A and B are symmetricn脳n matrices, the A+B must be symmetric as well.

It is given that A and B are symmetricn脳n matrices.

This means,AT=A,BT=B.

Here, the matrix A+B is symmetric, ifA+BT=A+B .

Check:

A+BT=AT+BT=A+B

So, A+BT=A+B.

This means, A+Bis also a symmetric matrix.

Then the given statement is true, by the properties of the transpose of matrices.

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