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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q65E (page 293)

Let $M_{n}$ be therole="math" localid="1660405225698" $n 脳 n$ matrix with l's on the main diagonal and directly above the main diagonal,role="math" localid="1660405220979" $- 1$ 's directly below the main diagonal, androle="math" localid="1660405229621" $0$ 's elsewhere. For example $M_{4} = [\begin{matrix} 1 & 1 & 0 & 0 \\ - 1 & 1 & 1 & 0 \\ 0 & - 1 & 1 & 1 \\ 0 & 0 & - 1 & 1 \end{matrix}]$ ,
Let $d_{n} = d e t (M_{n})$
a. For $n 猢� 3$ , find a formula expressing $d_{n}$ in terms of $d_{n - 1}$ and $d_{n - 2}$ .
b. Find $d_{1}, d_{2}, d_{3}, d_{4}$ , and $d_{10}$ .
c. For which positive integers $n$ is the matrix $M_{n}$ invertible?

Short Answer

Expert verified

Therefore,

a) $d_{n} = d_{n - 1} + d_{n - 2} .$

b) $d_{1} = 1, d_{2} = 2, d_{3} = 3, d_{4} = 5, d_{10} = 89$

c) $M_{n}$ is invertible for all $n 鈭� 鈩�$ .

Step by step solution

01

Step by Step Solution: Step 1: Matrix Definition

Matrix is aset of numbers arranged in rowsand 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� $m$ by $n$ 鈥� matrix, written 鈥� $m 脳 n$ .鈥�

02

(a)Step 2: To find the formula expressing in dn terms

Using the Laplace expansion along the first column, we have:

$d_{n} = 1 d_{n - 1} - (- 1) 路 1 路 d_{n - 2} d_{n} = d_{n - 1} + d_{n - 2} .$

03

(b)Step 3: To find d1,d2,d3,d4 and d10.

It applies that:

$d_{n} = d_{n - 1} + d_{n - 2} . d_{1} = 1, d_{2} = 2, d_{3} = 3, d_{4} = 5, d_{10} = 89 .$

04

(c)Step 4: To find which positive integers matrix Mn of is invertible

Given the formula for recursion $d_{n} = d_{n - 1} + d_{n - 2}$ , $M_{n}$ is invertible for all $n 鈭� 鈩�$ .

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Most popular questions from this chapter

Is the determinant of the matrix

$A = [\begin{matrix} 1 & 1000 & 2 & 3 & 4 \\ 5 & 6 & 7 & 1000 & 8 \\ 1000 & 9 & 8 & 7 & 6 \\ 5 & 4 & 3 & 2 & 1000 \\ 1 & 2 & 1000 & 3 & 4 \end{matrix}]$

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

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

5. $[\begin{matrix} 0 & 2 & 3 & 4 \\ 0 & 0 & 0 & 4 \\ 1 & 2 & 3 & 4 \\ 0 & 0 & 3 & 4 \end{matrix}]$

Find the determinant of the (2n) x (2n)matrix $A = [\begin{matrix} 0 & I_{n} \\ I_{n} & 0 \end{matrix}]$

In an economics text, $^{10}$ we find the following system:

$s Y + a r = I 掳 + G m Y - h r = M_{s} - M$

Solve for Y and r.

A square matrix is called a permutation matrix if each row and each column contains exactly one entry 1, with all other entries being 0 . Examples are $I_{n}, [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}]$ , and the matrices considered in Exercises 53 and 56 . What are the possible values of the determinant of a permutation matrix?

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