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Chapter 6: Q61E (page 293)

If the equation det A=det B holds for two nxn matrices A and B, is A necessarily similar to B?

Short Answer

Expert verified

Therefore, A and B are not similar.

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 ncolumns, the matrix is said to be a 鈥�m by n鈥� matrix, written 鈥� mxn.鈥�

02

To find if   is similar to  

Letn=2,A=4001,B=2002.Clearly,detA=detB=4.Assume,AandBaresimilar.Then,鈭�T鈭�鈩�2x2,T-1AT=BA=TBT-1A=2TIT-1A=2IA鈮�A.Wegotacontradiction.So,AandBarenotsimilar.

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Most popular questions from this chapter

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

21.T(f(t))=f(-t)fromP3toP3

Is the determinant of the matrix

A=[1100023456710008100098765432100012100034]

positive or negative? How can you tell? Do not use technology.

Consider two distinct real numbers, a and b. We define the function

f(t)=det[111abta2b2t2]

a. Show thatf(t) is a quadratic function. What is the coefficient oft2?
b. Explain whyf(a)=f(b)=0. Conclude thatf(t)=k(t-a)(t-b), for some constant k. Find k, using your work in part (a).
c. For which values of tis the matrix invertible?

For an invertible nxn matrix A, what is the relationship between det (A) and det (A)?

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