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Chapter 6: Q14E (page 309)

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

Short Answer

Expert verified

Therefore, the given statement is false.

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 an 鈥渕 by n 鈥� matrix, written 鈥� m脳n.鈥�

02

To check whether the given condition is true or false.

It applies that,

detAT=detA

detAT=1detA-1, Which is not necessarily equal to detA-1.

Therefore, the given statement is false.

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Most popular questions from this chapter

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

If the determinant of a2脳2 matrix is 4, then the inequality |Av鈫�|鈮�4|v鈫�| must hold for all vectors v鈫� in.

If twon脳n matrices A and B are similar, then the equationdetA=detB must hold.

Find the 3-volume of the 3-parallelepiped defined by the vectors

[1000],[1111],[1234].

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?
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