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Chapter 5: Q4E (page 263)

TRUE OR FALSE?The equation(AB)T=ATBT holds for all n脳n, matricesA andB.

Short Answer

Expert verified

The given statement is false

Step by step solution

01

Recollect the properties of the transpose of a matrix.

According to the properties of a transpose of a n脳nmatrix, for any Aand Bmatrices, (AB)T=BTAT.

02

Check whether the given statement is a true or false.

The given statement is The equation(AB)T=ATBT holds for alln脳n matricesA and B.

It is given that the vectors(AB)T=ATBT .

But, by the above stated property,(AB)T=BTAT for alln脳n matricesA and B.

It is know that the matrix multiplication is not commutative.

So, it is clear that ATBT=BTATis cannot be true for every case.

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

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Most popular questions from this chapter

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

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.

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

15.[21-2]

If then脳n matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?B-1.

Is there an orthogonal transformation T from R3to R3such thatT[230]=[302]

and T[230]=[302]?
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