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Chapter 6: Q39E (page 290)

IfAis an invertible matrix, what can you say about the sign ofdet(ATA)?

Short Answer

Expert verified

Therefore, the determinant of the matrix ATAis always positive.

Step by step solution

01

Step by Step Solution: Step 1: Definition

A determinant is a unique number associated with asquare matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

02

If det(ATA) is positive or negative

If Ais an invertible matrix, then:

detATA=detATdetAdetATA=detAdetAdetATA=(detA)2>0

Therefore, the determinant of the matrix is always positive.

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Most popular questions from this chapter

There exist real invertible 3脳3 matrices A andSsuch thatS-1AS=2A .

Question: Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

6.[111111441-12-21-18-8]

Find the determinants of the linear transformations in Exercises 17 through 28.

19.T(f)=2f+3f'fromP2toP2

Does the following matrix have an LU factorization? See Exercises 2.4.90 and 2.4.93.

A=[742531314]

det(A+B)=detA+detBfor all5脳5 matricesA andB.
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