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Chapter 5: Q12E (page 263)

TRUE OR FALSE? The equation det(AT)=det(A)holds for all 2脳2matrices A.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

 Step 1: Definition of the determinant.

Suppose A=[abcd]is a 2脳2matrix.

Then the determinant of matrix Ais defined as,

det(A)=[abcd]=ad-bc

02

 Step 2: Check whether the given statement is a true or false.

The given statement is the equation detAT=detAholds for all 2脳2matrix A.

Let A=abcd.

Then,

AT=abcdTAT=acbd

Find detAT,detA

detA=abcd=ad-bcdetAT=acbdc=ad-cd

This is clear that, the equation detAT=detAis holdsfor all2脳2 matrix A

Then the given statement is true.

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Most popular questions from this chapter

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

20.[235046007]

If A is an n脳nmatrix such that role="math" localid="1659514225617" AAT=In, then A must be an orthogonal matrix.

Question: If the n脳nmatrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?A+B.

If A=QRis a QRfactorization, what is the relationship between ATA and RTR?

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