In how many ways can 3 boys and 3 girls sit in a row if the boys and girls are each to sit together?

Answer:

The statements that are correct are:

c) [tex](in+a)(in+a^{-1})=2in+a+a^{-1}[/tex]

d) [tex]a^2b^7[/tex] is invertible.

e) [tex]a+b[/tex] is invertible.

Step-by-step explanation:

We are given that:

a and b are invertible n×n matrices.

We have to tell which of the following statements are true.

a)

[tex](ab)^{-1}=a^{-1}b^{-1}[/tex]

This statement is false.

Since:

[tex](ab)^{-1}=b^{-1}a^{-1}[/tex] and it may not be equal to the term [tex]a^{-1}b^{-1}[/tex]

b)

[tex]aba{-1}=b[/tex]

This expression could also be written as:

[tex]ab=ba[/tex]

Since on Post multiplying by a on both the sides.

But here we don't know whether the matrices are commutative or not.

Hence, the statement is false.

c)

[tex](in+a)(in+a^{-1})=2in+a+a^{-1}[/tex]

This statement is true.

since,

[tex](in+a)(in+a^{-1})=in(in+a^{-1})+a(in+a^{-1})\\\\=in^2+in.a^{-1}+a.in+aa^{-1}\\\\=in+a^{-1}+a+in\\\\=in+a^{-1}+a[/tex]

where in denote the identity matrix.

and we know that:

[tex]in^2=in[/tex]

d)

[tex]a^2b^7[/tex] is invertible.

This statement is true.

Since we know that prodct of invertible matrices is also invertible.

As [tex]a[/tex] is invertible so is [tex]a^2[/tex].

Also [tex]b[/tex] is invertible so is [tex]b^7[/tex].

Hence Product of  [tex]a^2[/tex] and  [tex]b^7[/tex] is also invertible.

i.e. [tex]a^2b^7[/tex] is invertible.

e)

[tex]a+b[/tex] is invertible.

This statement is true as sum of two invertible matrices is invertible.

f)

[tex](a+b).(a-b)=a^2-b^2[/tex]

This statement is false.

Since,

[tex](a+b).(a-b)=a(a-b)+b(a-b)\\\\=a.a-a.b+b.a-b.b\\\\=a^2-ab+ba-b^2[/tex]

Now as we are not given that:

[tex]ab=ba[/tex]

Hence, we could not say that:

[tex](a+b).(a-b)=a^2-b^2[/tex]

The correct statements which are true for all invertible [tex]\left({n\times n}\right)\cdot\left({n\times n}\right)[/tex] are:

(c). [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex]

(d). [tex]{a^2}{b^7}[/tex] is invertible.

(e). [tex]a+b[/tex] is invertible.

Further Explanation:

Given:

The matrix [tex]a[/tex] and [tex]b[/tex] are invertible [tex]\left({n\times n}\right)[/tex] matrices.

Calculation:

(a)

The statement is [tex]{\left({ab}\right)^{-1}}={a^{-1}}{b^{-1}}[/tex] false.

[tex]\begin{aligned}{\left({ab}\right)^{-1}}&={a^{-1}}{b^{-1}}\\\left({ab}\right){\left({ab}\right)^{-1}}&=\left({ab}\right){a^{-1}}{b^{-1}}\\I&\ne ab{a^{-1}}{b^{-1}}\\\end{aligned}[/tex]

The statement is [tex]{\left({ab}\right)^{-1}}={a^{-1}}{b^{-1}}[/tex] false.

(b)

The statement is [tex]ab{a^{-1}}=b[/tex].

Now multiply by a both the side.

[tex]\begin{aligned}ab{a^{-1}}a&=ba\\ab&\ne ba\\\end{aligned}[/tex]

The statement is [tex]ab{a^{-1}}=b[/tex] is false.

(c)

The statement is [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex].

Solve the above equation to check whether it is invertible.

[tex]\begin{aligned}\left({in+a}\right)\left({in+{a^{-1}}}\right)&=in\left({in+{a^{-1}}}\right)+a\left({in+{a^{-1}}}\right)\\&=i{n^2}+in\cdot{a^{-1}}+a\cdot in+a{a^{-1}}\\&=in+{a^{-1}}+a+in\\&=in+{a^{-1}}+a\\\end{aligned}[/tex]

The statement is true.

(d)

The statement is [tex]{a^2}{b^7}[/tex].

The product of invertible matrices is always invertible.

As [tex]a[/tex] is invertible so [tex]{a^2}[/tex] is also invertible.

As [tex]b[/tex] is invertible so [tex]{b^7}[/tex] is also invertible.

Hence, the product of [tex]{a^2}[/tex] and [tex]{b^7}[/tex] is also invertible.

The statement is true.

(e)

The statement [tex]a+b[/tex] is true as [tex]a+b[/tex] is always invertible.

(f)

The statement is [tex]\left({a+b}\right)\cdot\left({a-b}\right)={a^2}-{b^2}[/tex].

Solve the equation to check the inevitability.

[tex]\left({a+b}\right)\times\left({a-b}\right)={a^2}-ab+ba-{b^2}[/tex]

The statement is not true as [tex]ab=ba[/tex].

Hence, the correct statements which are true for all invertible [tex]\left({n\times n}\right)\cdot\left({n\times n}\right)[/tex] are:

(c). [tex]\left({in+a}\right)\left({in+{a^{-1}}}\right)=2in+a+{a^{-1}}[/tex]

(d). [tex]{a^2}{b^7}[/tex] is invertible.

(e). [tex]a+b[/tex] is invertible.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Linear equation

Keywords: Invertible, matrices, matrix, statement, function, true, determinants, elements, inverse.

To determine how many 5/16 are in 1 7/8, you convert 1 7/8 to an improper fraction, 15/8, and then divide it by 5/16, which yields 6.

To find out how many 5/16 are in 1 7/8, you first need to convert 1 7/8 to a improper fraction. Since 1 is equal to 8/8, adding 7/8 gives us 15/8. Now, to find how many 5/16 are in 15/8, you can divide 15/8 by 5/16. To do this, you multiply 15/8 by the reciprocal of 5/16, which is 16/5.

15/8 × 16/5 = (15×16)/(8×5) = 240/40. This simplifies to 6, meaning there are 6 segments of 5/16 in 1 7/8.

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Answer:

The inverse statement is

"A number that does not has exactly two distinct factors is not prime"

Step-by-step explanation:

Given a statement we have to tell the inverse of that statement.

The given statement is

"A number that has exactly two distinct factors is prime".

We have to write the inverse of the above statement.

The inverse of the statement is

"A number that does not has exactly two distinct factors is not prime"

1 gallon = 8 pints

15/8 = 1.875 gallons per minute

60 minutes per hour

1.875 * 60 = 112.5 gallons per hour

The symbol μ is used for population mean, while x is used for sample mean in statistics.

The symbol μ (pronounced "mew") is used to represent the population mean.

The population mean is a parameter that represents the average value of a variable in the entire population.

On the other hand, the symbol x is used to represent the sample mean, which is the average value calculated from a subset of the population called a sample.

The symbol μ (pronounced "mew") is used to represent the population mean.

It denotes the mean of all values within an entire population, which is a parameter of that population.

To illustrate why μ represents the population mean, consider a dataset: 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4.

If this were to represent an entire population, we would calculate the mean and then denote it with μ.

Answer with explanation:

Smallest Polygon = Triangle , which has 3 sides

Sum of angles of triangle = 180°→→, Using the angle sum property of triangle.

We can find the Sum of angles of triangle, using the formula

   = (n-2)×180°

  = (3-2)×180°

 = 1 × 180°

 = 180°

A Quadrilateral has four sides.

Sum of angles of Quadrilateral = 360°

Or , using the formula

=(4-2)×180°

=2 × 180°

=360°

For, n sided Polygon

Sum of it's Interior angles = (n-2) × 180°

To prove this, we should divide that polygon into triangles, and multiply number of triangles by angle of 180°.

For, example, Quadrilateral can be divided into two triangles , sum of interior angles of Quadrilateral =2 × 180°=360°

Answer: He traveled 54.28 miles per hour as an average.

Step-by-step explanation:

Since we have given that

Number of miles traveled in first hour = 52 miles

Number of miles traveled in second hour = 64  miles

Number of miles traveled in third hour = 49 miles

Average per hour is given by

[tex]=\dfrac{3}{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}\\\\\\=\dfrac{3}{\dfrac{1}{52}+\dfrac{1}{64}+\dfrac{1}{49}}\\\\\\=\dfrac{3}{\dfrac{784+637+832}{40768}}\\\\\\=\dfrac{3}{\dfrac{2253}{40768}}\\\\\\=\dfrac{40768\times 3}{2253}\\\\\\=\dfrac{122304}{2253}\\\\\=54.28\ miles/hr[/tex]

Hence, he traveled 54.28 miles per hour as an average.

Answer:-9 degrees

Step-by-step explanation:The average drop in temperature per minute is 9 degrees.

for a, make a dot on the line plot at each of the given lengths

b) there are more fishes greater than 1 1/2

Answer:

[tex]2 + \frac{2}{10}+\frac{2}{100}[/tex]

Step-by-step explanation:

In the expanded form we write a number as the sum of all place value of its digits from left to right,

Here, the given number,

2.22

By the place value chart,

Place value of first digit = 2 ( ones )

Place value of the first digit after decimal = [tex]\frac{2}{10}[/tex] ( tenth )

Place value of the second digit after decimal = [tex]\frac{2}{100}[/tex] ( hundredth )

[tex]\implies 2.22 = 2 + \frac{2}{10}+\frac{2}{100}[/tex]

The expected net winnings are  -$36. That is you lose $36 according to chance.

We have given

A game of chance involves rolling a 14-sided die once. If a number from 1 to 3 comes up, you win 2 dollars.

We have to determine expected net winnings

The expected value is what the player can expect to win or lose if they were to play many times with the same game.

let's say that you have a 3/13 chance of winning $3 and a 2/13 chance of winning $10 and an 8/13 chance of winning $0. 

Therefore if the dice is rolled 13 times then the expected to win 3X$3 + 2X$10 = $29 

But you pay $5 per roll so 13 rolls would cost me 13X$5 = $65 

Therefore the expected net winnings are $29-$65 = -$36 

Therefore the expected net winnings are  -$36.

So you lose $36 according to chance.

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Final answer:

The graph of f(x) = 1/x is a hyperbola with vertical and horizontal asymptotes at x = 0 and y = 0, respectively. The domain is all real numbers except 0, and the range is the same. Unlike linear, quadratic, or exponential functions, the hyperbolic shape of 1/x presents its unique characteristics needing careful domain consideration.

Explanation:

The graph of the basic rational function f(x) = 1/x is characterized by two distinct branches, one in the first quadrant (where x and y are both positive) and the other in the third quadrant (where x and y are both negative). This function is a hyperbola, and as the value of x approaches zero from either the positive or negative side, the value of f(x) increases without bound, referred to as a vertical asymptote at x = 0. Additionally, as the value of x becomes very large or very small (approaching positive or negative infinity), the value of f(x) approaches zero, creating a horizontal asymptote at y = 0. Importantly, this function is undefined at x = 0, so the domain is all real numbers except 0, (domain: ℝ \ {0}). The function takes on all real number values for the output, so the range is also all real numbers except 0 (range: ℝ \ {0}).

Comparing to other types of functions, unlike linear functions like f(x) = 7x which produces a straight line or quadratic functions such as f(x) = 2x² which graphs as a parabola, the rational function f(x) = 1/x leads to a non-linear graph with two separate branches. Exponential functions like f(x) = eⁿ¹, however, grow or decline at an increasing rate, which is a different behavior compared to the hyperbolic decline or rise of a rational function. It's vital to note that as with the function |√x|, there are points along the graph of 1/x where the function is not defined, highlighting the importance of considering the domain in function analysis.

The function 1/x² also represents a rational function but with a different characteristic as it exhibits positive blowup on both sides. This means that as x approaches zero, whether from the positive or negative side, the function tends to positive infinity, a behavior different from the reciprocal function 1/x.