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Chapter 6: Q25E (page 309)

25. If the determinant of a square matrix is -1, then Amust be an orthogonal matrix.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Orthogonal Matrix Definition 

A square matrix with real numbers or elements is said to be an orthogonal matrix, if its transpose is equal to its inverse matrix.

Or we can say, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

02

To check whether the given condition is true or false

If the determinant of a square matrix is -1, thenmust be an orthogonal matrix or not.

For example,

A=12002

detA=2-12-0

detA=-1

A is not an orthogonal matrix, yet det A = - 1.

Therefore, the given condition is false.

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Most popular questions from this chapter

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]

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

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

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).

Question: Let A and B be 2 x 2 matrices with integer entries such that A,A+B,A+2B,A+3B, and A+4B are all invertible matrices whose inverses have integer entries. Show that A+5B is invertible and that it鈥檚 inverse has integer entries. This question was in the William Lowell Putnam Mathematical Competition in 1994. Hint: Consider the function F(t)=(det(A+tB))2 -1. Shows that this is a polynomial; what can you say about its degree? Find the values f(0), f(1), f(2), f(3), f(4),using Exercise 53. Now you can determine f(t) by using a familiar result: If a polynomial f(t) of degree <m has more than m zeros, then f(t)=0 for all t.

Which of the following functions F of A=[abcd] are linear in both columns? Which are linear in both rows? Which are alternating on the columns?
role="math" localid="1659502796300" a.F(A)=bcb.F(A)=cdc.F(A)=acd.F(A)=bc-ade.F(A)=c
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