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Chapter 6: Q4E (page 308)

If matrixB is obtained by swapping two rows of a n脳nmatrix A, then the equationdetB=-detA must hold.

Short Answer

Expert verified

Therefore, the given condition is true.

Step by step solution

01

Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be a 鈥� mby n鈥� matrix, written 鈥� m脳n.鈥�

02

To check whether the given condition is true or false

We can see that this also applies from the definition.

If Bis obtained from by Aswitching the i-th and j-th row, and an added

(-1)I(p)鈭�k=1nak,p(k)

By calculating detA, corresponds to a permutation p鈭�Sn, then the same added while calculating detB will correspond to the permutation where p(i)and p(j)switch places, so the addend will change its sign.

Thus,

detB=-detA

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Most popular questions from this chapter

If is any noninvertible square matrix, then detA=det(rrefA).

IfA is a 4脳4 matrix whose entries are all 1 or -1 , then detAmust be divisible by 8 (i.e., detA=8kfor some integer k).

Consider a 4x4 matrix A with rows V1鈫�,V2鈫�,V3鈫�,V4鈫�. If det(A) = 8, find the determinants in Exercises 11 through 16.

16. role="math" localid="1659506283449" det[6v鈬赌1+2v鈬赌4v鈬赌2v鈬赌33v鈬赌1+v鈬赌4]

In an economics text,10we find the following system:

sY+ar=I掳+GmY-hr=Ms-M

Solve for Y and r.

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

7. [0000100012001230123412345]
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