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morpeh [17]
3 years ago
7

An n × n matrix B has characteristic polynomial p(λ) = −λ(λ − 3) 3 (λ − 2) 2 (λ + 1). Which of the following statements is false

? (a) rank B = 6. (b) det (B) = 0. (c) det (B TB) = 0. (d) B is invertible. (e) n = 7
Mathematics
1 answer:
asambeis [7]3 years ago
8 0

Answer:

Only d) is false.

Step-by-step explanation:

Let p=p(\lambda)=\lambda(\lambda-3)^3 (\lambda-2)^2 (\lambda+1) be the characteristic polynomial of B.

a) We use the rank-nullity theorem. First, note that 0 is an eigenvalue of algebraic multiplicity 1. The null space of B is equal to the eigenspace generated by 0. The dimension of this space is the geometric multiplicity of 0, which can't exceed the algebraic multiplicity. Then Nul(B)≤1. It can't happen that Nul(B)=0, because eigenspaces have positive dimension, therfore Nul(B)=1 and by the rank-nullity theorem, rank(B)=7-nul(B)=6 (B has size 7, see part e)

b) Remember that p(\lambda)=\det(B-\lambda I). 0 is a root of p, so we have that p(0)=\det(B-0 I)=\det B=0.

c) The matrix T must be a nxn matrix so that the product BTB is well defined. Therefore det(T) is defined and by part c) we have that det(BTB)=det(B)det(T)det(B)=0.

d) det(B)=0 by part c) so B is not invertible.

e) The degree of the characteristic polynomial p is equal to the size of the matrix B. Summing the multiplicities of each root, p has degree 7, therefore the size of B is n=7.      

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Answer:

7 . 4 + 6 - 12 : 4 = 31

Step-by-step explanation:

* To solve this problem lets revise the order of operations in

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# Addition

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# Multiplication

# Division

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# Grouping ⇒ Parenthesis or brackets

- The order of these operations is:

# Parenthesis

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# Multiplication and Division which comes first from left to right

# Addition and Subtraction which comes first from left to right

- There is a word made from the first letter of each operation

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* Lets solve the problem

∵ 7 . 4 + 6 - 12 : 4

∵ The (.) means multiply the numbers

∵ The symbol (:) means divided the numbers

∴ At first multiply 7 by 4 and divide 12 by 4

∴ (7 × 4) + 6 - (12 ÷ 4)

∴ 28 + 6 - 3

∵ Addition comes before subtraction from the left

∴ (28 + 6) - 3

∴ 34 - 3 = 31

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Let a1, a2, a3, ... be a sequence of positive integers in arithmetic progression with common difference
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a_3 = a_2 + 2 = a_1 + 2\cdot2

a_4 = a_3+2 = a_1+3\cdot2

\cdots \implies a_n = a_1 + 2(n-1)

and since b_1,b_2,b_3,\cdots are in geometric progression,

b_2 = 2b_1

b_3=2b_2 = 2^2 b_1

b_4=2b_3=2^3b_1

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Recall that

\displaystyle \sum_{k=1}^n 1 = \underbrace{1+1+1+\cdots+1}_{n\,\rm times} = n

\displaystyle \sum_{k=1}^n k = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2

It follows that

a_1 + a_2 + \cdots + a_n = \displaystyle \sum_{k=1}^n (a_1 + 2(k-1)) \\\\ ~~~~~~~~ = a_1 \sum_{k=1}^n 1 + 2 \sum_{k=1}^n (k-1) \\\\ ~~~~~~~~ = a_1 n +  n(n-1)

so the left side is

2(a_1+a_2+\cdots+a_n) = 2c n + 2n(n-1) = 2n^2 + 2(c-1)n

Also recall that

\displaystyle \sum_{k=1}^n ar^{k-1} = \frac{a(1-r^n)}{1-r}

so that the right side is

b_1 + b_2 + \cdots + b_n = \displaystyle \sum_{k=1}^n 2^{k-1}b_1 = c(2^n-1)

Solve for c.

2n^2 + 2(c-1)n = c(2^n-1) \implies c = \dfrac{2n^2 - 2n}{2^n - 2n - 1} = \dfrac{2n(n-1)}{2^n - 2n - 1}

Now, the numerator increases more slowly than the denominator, since

\dfrac{d}{dn}(2n(n-1)) = 4n - 2

\dfrac{d}{dn} (2^n-2n-1) = \ln(2)\cdot2^n - 2

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2^n > \dfrac4{\ln(2)} n \implies \ln(2)\cdot2^n - 2 > 4n - 2

This means we only need to check if the claim is true for any n\in\{1,2,3,4\}.

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If n=2, then

c = \dfrac{4}{2^2 - 4 - 1} = \dfrac4{-1} = -4 < 0

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c = \dfrac{12}{2^3 - 6 - 1} = 12

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There is only one value for which the claim is true, c=12.

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Total distance from the moon = Number of paper clips used × Length of one clip

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Distance from the moon is 2.4 × 10⁵ miles.

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