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Chapter 6: Q41E (page 291)

Consider a skew-symmetricn脳n matrix A, where n is odd. Show that A is noninvertible, by showing that detA=0.

Short Answer

Expert verified

It is proved thatdetA=0.

Step by step solution

01

Step by Step Solution: Step 1: 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 mrows and ncolumns, the matrix is said to be a 鈥� mby n鈥� matrix, written 鈥�m脳n.鈥�

02

To show that detA=0.

Given that Ais skew-symmetric.

Hence,

At=-A

Taking determinants we get,

detAt=det(-A)

Now we know that detAt=detAand role="math" localid="1660380315558" det(-A)=(-1)n.

Hence, we get detA=(-1)ndetA=-detA(as is odd.)

This implies,

2detA=0detA=0

Hence, it鈥檚 proved that detA=0.

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Most popular questions from this chapter

If Ais an invertible n脳n matrix, thendet(AT) must equaldet(A-1).

If A and Bare invertible n脳nmatrices, and if Ais similar toB, isadj(A) necessarily similar to adj(B)?

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

25. T(M)=[2304]M from the space V of upper triangular matrices to V

a. Find a noninvertible 2脳2matrix whose entries are four distinct prime numbers, or explain why no such matrix exists.

b. Find a noninvertible 3脳3matrix whose entries are nine distinct prime numbers, or explain why no such matrix exists.

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