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Chapter 5: Q14E (page 263)

Question:TRUE OR FALSE?If A and Bare symmetricn脳n matrices,AB then must be symmetric as well.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Definition of a symmetric matrix.

Suppose A is ann脳nmatrix.

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

By the properties of transpose,(AB)T=BTAT

02

Check whether the given statement is a true or false.

The given statement is if A and B are symmetric n脳nmatrices, the AB must be symmetric as well.

It is given that A and B are symmetric n脳nmatrices.

This means,AT=A,BT=B

Here, the matrix AB is symmetric, ifABT=AB

Check:

ABT=ATBT=BA

So,ABT=AB

It is know that the matrix multiplication is not commutative.

So, it is clear that AB = BA is cannot be true for every case.

So,ABT鈮�AB

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

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Most popular questions from this chapter

If the nxn matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? 3A.

Find the angle胃 between each of the pairs of vectorsu鈫� and in exercises 4 through 6.

5.u鈫�=[123V鈫�=234] .

Question: Consider an n脳n matrix A. Show that A is an orthogonal matrix if (and only if) A preserve the dot product, meaning that(Ax鈫�)=.(Ay鈫�)=x鈫�.y鈫� for allrole="math" localid="1659499729556" x鈫� andy鈫� in Rn.

If is a symmetric matrix, then must be symmetric as well.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

4路[403],[250-25],[0-20]
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