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Chapter 6: Q16E (page 309)

There exists a nonzero4脳4 matrixAsuch thatdetA=detA(4A).

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

02

To check whether the given condition is true or false.

For example,

A=0001000000000000


Indeed,

detA=0=det(4A).

Therefore, the given condition is true.

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Most popular questions from this chapter

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

5.[0234000412340034]

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.

Even if an n脳n matrix A fails to be invertible, we can define the adjoint adj(A)as in Theorem 6.3.9. The ijthentry of adj(A)is (-1)i+jdet(Aji). For which n脳nmatrices A isadj(A)=0 ? Give your answer in terms of the rank ofA. See Exercise 41.

Question:A basisv1鈫�,v2鈫�,v3鈫�of鈩�3is called positively oriented ifv1鈫�encloses an acute angle withv2鈫�脳v3鈫�. Illustrate this definition with a sketch. Show that the basis is positively oriented if (and only if)det[v1鈫�,v2鈫�,v3鈫�]is positive.

Demonstrate Theorem 6.3.6 for linearly dependent vectorv鈬赌1,....,v鈬赌m.
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