Mathematics High School

proof let a be a square matrix of order n. (a)show that is symmetric. (b)show that is skew-symmetric. (c)prove that a can be written as the sum of a symmetric matrix b and a skew-symmetric matrix c, a

Answers

Answer 1

(a)Symmetric matrix -

Let matrix A be of order 3×3 =

[tex]\left[\begin{array}{ccc}1&3&8\\3&8&-4\\8&-4&6\end{array}\right][/tex]

So, the transpose of A will be -

A' =

[tex]\left[\begin{array}{ccc}1&3&8\\3&8&-4\\8&-4&6\end{array}\right][/tex]

So, we can say that A=A'. So, it is symmetric.

(b) Skew - symmetric matrix -

Let us consider a matrix B =

[tex]\left[\begin{array}{ccc}0&-6&4\\-6&0&7\\4&7&0\end{array}\right][/tex]

matrix B is said to be skew-symmetric if aij =−aji.

(B' =−B)

B' =

[tex]\left[\begin{array}{ccc}0&6&-4\\6&0&-7\\-4&-7&0\end{array}\right][/tex]

(c) A + A′ is a symmetric matrix for any square matrix A with real number entries, whereas A - A′ is a skew-symmetric matrix.

Proof: If B = A + A', then B ′ = (A + A ) ′

(A + B)′ = A′ + B′, thus = A′ + (A′)′

(as (A′)′ =A) = A′ + A

= B (because B + A = B + A′)

B = A+A′ is a symmetric matrix as a result.

Let C now equal A-A′.

Why does C' equal (A-A′)′=(A′)′?

= A′ - A (Why?)

=- (A - A′) = - C

A - A′ is a skew-symmetric matrix as a result.

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Related Questions

The graph shows the cost, c, in dollars of using a carpooling
service for d days. Which equation represents the cost of the carpooling service?
A c = = d
B c = 6d
C d = 6c
D c = 6

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Answer: where is the graph?

Name the geometric term modeled by the given object.
a telephone pole

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The geometric term modeled by the given object (a telephone pole) is a line

How to determine the geometric term?

The given object is a telephone pole

The pole of a telephone pole is often straight.

This means that the telephone pole can be modelled by a line segment

Hence, the geometric term modeled by the given object (a telephone pole) is a line

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The sequence of transformations could be applied to the parent function, f(x) = x, to obtain the graph is E. Shift up 6 units, reflect over the x-axis, and then vertically stretch by a factor of 2.

What is transformation?

It should be noted that transformation simply means the movement of an object from a position to another position.

In this case, the sequence of transformations could be applied to the parent function, f(x) = x, to obtain the graph is to shift up 6 units, reflect over the x-axis, and then vertically stretch by a factor of 2.

Therefore sequence of transformations could be applied to the parent function, f(x) = x, to obtain the graph is E. Shift up 6 units, reflect over the x-axis, and then vertically stretch by a factor of 2.

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1. Turn this standard notation 25,000,000 into scientific notation.

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Answer:

25,000,000 = 2.5 × 10^7

The ^ is raised to the power.

Hope this helps :)

Step-by-step explanation:

2.5 × 10^7 = 25,000,000

there are 11 books on a shelf. 9 of these books are new. the rest of them are used

Answers

Two of them are used and 9 of them are not

Where on the number line would you plot the square roots of the numbers 70, 85, 99, and 112?
a. 4
b. 5
c. between 4 and 5
d. between 64 and 125

Answers

We have to plot the square roots of the numbers 70, 85, 99, and 112 between 64 and 125 ( d )

First, we must evaluate the square root of given numbers.

√70 = √ ( 2 × 5 ×7 ) = 8.36

√85 = √ ( 5 × 17 ) = 9.22

√99 = √( 9 × 11 ) = 3√11 = 9.95

√112 = √( 4 × 4 × 7 ) = 4 √7 = 10.58

All these numbers ( 8.36, 9.22, 9.95 , 10.58 ) belong between, square roots of √64 and √125 which are 8 and 15. So we have to put these numbers in between the square roots of 64 and 125 on the number table.

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Evaluate the expression when x=3 , y=−4 , and z=−6.
z−2xy=

Answers

x = 3 , y = −4 , and z = −6

z-2xy =

Substitute the value of x,y and z in the expression.

-6-2(3)(-4)

-6-2(-12)

-6+24

5 1-3a

Note: Please make sure to properly format your answers. All dollar figures in the answers need to include the dollar sign and any amount over 1,000 should include the comma
($2,354.67). All percentage values in the answers need to include a percentage sign (%). For all items without specific rounding instructions, round your answers to two decimal
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The variance of a distribution is 195. What is the standard deviation, rounded to the nearest thousandth?

Answers

Applying the square root of the variance, the standard deviation of the distribution is of 13.964.

What is the standard deviation of a distribution?

The standard deviation of a distribution is given by the square root of the variance of the distribution.

For this problem, the variance of the distribution is of 195, hence the standard deviation is:

s = sqrt(195) = 13.964.

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a certain composer wrote 10 symphonies, 4 piano concertos (music for piano and orchestra), and 35 piano sonatas (music for solo piano). a. how many ways are there to play first a symphony and and then a piano concerto? b. the manager of a radio station decides that on each successive evening (7 days a week), a symphony will be played followed by a piano concerto followed by a piano sonata. for how many nights could this be carried out before exactly the same program would have to be repeated? about how many years would this be? (round to the nearest year).

Answers

(a) According to the question, a composer wrote [tex]10[/tex] symphonies, [tex]4[/tex] piano concerts (piano and orchestra) and [tex]35[/tex] piano sonatas (solo piano):

Here, determine how many ways are there to play first a symphony and then piano concerto.

So, the probability is [tex](10P1) (4P1) = (10)(4)= 40[/tex]

Therefore, there are [tex]40[/tex] ways to play first a symphony and then piano concerto.

(b)The manager of a radio station decides that on each successive evening, a symphony will be played followed by a piano concerto followed by a sonata.

Here determine how many nights could be carried out before the same program would get repeated.

So, the probability would be [tex](35)(10)(4)=1440[/tex]

Therefore, [tex]1440[/tex] nights are carried out before the same program gets repeated.

Now, to calculate the years: [tex]\frac{1440 nights}{365days} = 3.95 years[/tex]

Therefore, it would be [tex]3.95[/tex] years.

What is probability?

Probability is the method to measure the occurred events or the occurrence of the events. It is also used to analyze the outcomes of random events. For example, to calculate the probability of a playing card which is 52 in numbers. And it is divided into four cards and each are in the count of 13.These four cards are heart, spades, diamond and club.

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How many 5-letter words can be made from the letters WASTEFUL if

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Answers

Answer w a s t e f u l

Step-by-step explanation:

theres 7 words on it

suppose that a function f has domain (-2,2) and range (-3,5). what is the domain of the function f(2x)?

Answers

The domain of the function f(2x) is (-1, 1).

Given that a function f has domain (-2,2) and range (-3,5).

We need to determine the domain of the function f(2x).

To determine the domain of the function f(2x), we need to consider how the original domain (-2,2) is affected by the transformation of multiplying the input by 2.

When we multiply x by 2, it stretches or compresses the function horizontally.

In this case, multiplying by 2 will stretch the original domain by a factor of 1/2.

To find the new domain, we divide the original domain endpoints by 2:

New domain = (-2/2, 2/2) = (-1, 1)

Therefore, the domain of the function f(2x) is (-1, 1).

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Gogglejaws can eat 36 fish in one minute.
How many fish could Gogglejaws eat in 2.2
hours?

Answers

Answer:

3752

Step-by-step explanation:

there are 60 minutes in an hour so multiply 2.2 times 60

2.2 • 60 = 132 minutes

132 min • 36 fish = 3752

Yo help please I need this done it's about rational or irrational numbers u have to figure out if it's either rat or irr. If it's rational then figure out if it is a integers. If it's a integers then figure out if it's a whole number if it is then figure out if it's a natural number. Here's the order of it. R I W N and if it's irr then it's just I