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Chapter 6: Q 6.2-66E (page 293)

Question: Let Mn be the matrix with all 1's along the main diagonal, directly above the main diagonal, and directly below the diagonal, and 0's everywhere else.

For example,

Let dn = det ( Mn).
a. Find a formula expressing dn in terms of dn-1 and dn-2, for positive integers n>3.
b. Find d1,d2,,.....,d8.
c. What is the relationship between dn and dn+3? What about dn and dn+6 ?
d. Find d100.

Short Answer

Expert verified

Therefore,

Step by step solution

01

Matrix Definition 

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

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

02

To find the formula expressing in  dn, dn-1, dn-2 terms

Using the Laplace expansion along the first column, we have

03

To find d1,d2,,.....,d8..

It applies:

04

To find relationship between dn  and  dn+3

From part b,

We can see that dn+3= dn

Therefore, also dn+6= dn .

05

To find  d100

From part c,

We conclude that,

d100 = d4

d100 = -1

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Most popular questions from this chapter

There exist real invertible 3脳3 matrices A and S such that .

Consider a 4脳4matrix A with rowsv鈫�1,v鈫�2,v鈫�3,v鈫�4. If, find the determinants in Exercises 11 through16.

12. localid="1659509477853" det[v鈫�1v鈫�2v鈫�3v鈫�4]

(For those who have studied multivariable calculus.) Let Tbe an invertible linear transformation from鈩�2to鈩�2, represented by the matrix M. Let惟1be the unit square in 鈩�2and惟2its image under T . Consider a continuous functionf(x,y)from鈩�2to鈩�, and define the functiong(u,v)=f(T(u,v)). What is the relationship between the following two double integrals?

鈭�惟2f(x,y)dAand鈭�惟1g(u,v)dA

Your answer will involve the matrix M. Hint: What happens whenf(x,y)=1, for allx,y?

If all the entries of an invertible matrixA are integers, then the entries of A-1must be integers as well.

Use Exercise 31 to find

det[1111112345149162518276412511681256625]

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