The differential equation (x + 2y)dx +ydy = 0 can be solved using the substitution. Select the correct answer. a. U=x+2yb. U=yc. U=xyd. U=y/xe. It cannot be solved using a substitution

The probability of x being greater than 5 is approximately 0.3679.

To compute p(x > 5) for x with an exp(0.2) distribution, we can use the probability density function (PDF) of the exponential distribution:

f(x) = 0.2e^(-0.2x)

The probability of x being greater than 5 is given by the integral of the PDF from 5 to infinity:

p(x > 5) = integral from 5 to infinity of f(x) dx

= integral from 5 to infinity of 0.2e^(-0.2x) dx

= [-e^(-0.2x)] from 5 to infinity

= e⁻¹ˣ

= 0.3679

Therefore, the probability of x being greater than 5 is approximately 0.3679.

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The formula for the general term a_n of the given sequence. The sequence provided is: 3, 8, 13, 18, ...

Step 1: Identify the pattern
We can see that the difference between consecutive terms is constant:
8 - 3 = 5
13 - 8 = 5
18 - 13 = 5

Step 2: Define the sequence
Since the difference between consecutive terms is constant, this is an arithmetic sequence. The common difference (d) is 5.

Step 3: Find the formula for the general term a_n
The formula for the general term of an arithmetic sequence is:
a_n = a_1 + (n - 1) * d

where a_n is the nth term, a_1 is the first term, n is the position of the term in the sequence, and d is the common difference.

Step 4: Plug in the known values
In our case, a_1 = 3 and d = 5. Plugging these values into the formula, we get:
a_n = 3 + (n - 1) * 5

Step 5: Simplify the formula
a_n = 3 + 5n - 5
a_n = 5n - 2

So the formula for the general term a_n of the sequence is:
a_n = 5n - 2

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(1) To find the resonant frequencies of the model T(s) = 7/s(s2 +6s+58), we first need to factor the denominator:

s(s2 +6s+58) = s(s+3-√31i)(s+3+√31i)

The resonant frequencies occur at the poles of the transfer function, which are the roots of the denominator. Therefore, the resonant frequencies are:

ω1 = 0 (from the pole at s = 0)

ω2 = √31 (from the poles at s = -3±√31i)

(2) To find the resonant frequencies of the model T(s) = 7/ (3s2 +18s+174)(2s2 +85+58), we first need to factor the denominator:

(3s2 +18s+174)(2s2 +85+58) = 6(s+3+i√11)(s+3-i√11)(s+(-7+i√85)/2)(s+(-7-i√85)/2)

The resonant frequencies occur at the poles of the transfer function, which are the roots of the denominator. Therefore, the resonant frequencies are:

ω1 = √11 (from the poles at s = -3±i√11)

ω2 = √85/2 (from the poles at s = (-7±i√85)/2)

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The quotient (h(x+3))/(h(x)) is 125. This tells us that the value of h changes by a factor of 125 when the input is increased by 3.

To find the quotient (h(x+3))/(h(x)), we will first evaluate the function h(x) for the given inputs and then divide the two results.

The function h is given by h(x) = 5^(x).

1. Evaluate h(x+3): h(x+3) = 5^(x+3)
2. Evaluate h(x): h(x) = 5^x
3. Find the quotient: (h(x+3))/(h(x)) = (5^(x+3))/(5^x)

Using the properties of exponents, we can simplify the expression further:
(5^(x+3))/(5^x) = 5^(x+3-x) = 5^3 = 125

The quotient (h(x+3))/(h(x)) is 125. This tells us that the value of h changes by a factor of 125 when the input is increased by 3.

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Connect the endpoints of each new line segment to create the scaled polygon a.

To draw a scaled copy of polygon a using a scale factor of 2, follow these steps:

Choose a point on the paper that will be the centre of your scaling transformation.

Draw a line from the centre point to each vertex of the original polygon a.

Measure the length of each line segment.

Multiply each length measurement by a scale factor of 2.

From the centre point, draw a new line for each scaled segment with the new, scaled length.

Connect the endpoints of each new line segment to create the scaled polygon a.

Remember to label your scaled polygon a to indicate that it is a scaled copy and to note the scale factor used.

Complete Question:

Here are 3 polygons.

(Below mentioned diagram)

a) Draw a scaled copy of polygon a suing a scale factors of 2.

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To find the tangent line approximation for 1 ‾‾‾‾‾√ near =2, we first need to find the derivative of the function f(x) = 1 ‾‾‾‾‾√.

Using the power rule of differentiation, we get: f'(x) = 1/2(x)^(-1/2) Now, we can substitute the value a = 2 and f'(a) = f'(2) = 1/2(2)^(-1/2) = 1/2√2 into the formula: f(x) = f^1(a)(x-a) + f(a) to get the equation of the tangent line at x = 2: y = 1/2√2(x-2) + 1 Therefore, the tangent line approximation for 1 ‾‾‾‾‾√ near =2 is y = 1/2√2(x-2) + 1,

where the slope of the line is given by f'(2) and the point (2,1) lies on the line.  Use the formula for the tangent line approximation: f(x) ≈ f^1(a)(x-a) + f(a). For x near 2, f(x) ≈ (1/2√2)(x-2) + √2. This is the tangent line approximation for f(x) = √x near x = 2.

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The probability that all five balls drawn out of the box at random are white is approximately 0.001082.

To find the probability that all five balls drawn out of the box at random are white, follow these steps:

1. Calculate the total number of balls in the box: 5 white balls + 6 black balls = 11 balls
2. Determine the probability of drawing the first white ball: 5 white balls / 11 total balls = 5/11
3. After drawing the first white ball, there are now 4 white balls and 10 total balls remaining. Determine the probability of drawing the second white ball: 4 white balls / 10 total balls = 4/10
4. After drawing the second white ball, there are now 3 white balls and 9 total balls remaining. Determine the probability of drawing the third white ball: 3 white balls / 9 total balls = 1/3
5. After drawing the third white ball, there are now 2 white balls and 8 total balls remaining. Determine the probability of drawing the fourth white ball: 2 white balls / 8 total balls = 1/4
6. After drawing the fourth white ball, there is now 1 white ball and 7 total balls remaining. Determine the probability of drawing the fifth white ball: 1 white ball / 7 total balls = 1/7

To find the probability of all five events occurring, multiply the probabilities together: (5/11) * (4/10) * (1/3) * (1/4) * (1/7) = 0.00108225108

So, the probability that all five balls drawn out of the box at random are white is approximately 0.001082, or 0.1082%.

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To evaluate the definite integral ∫ 7 3 ( 5 x − 20 ) d x ∫37(5x-20)dx in terms of areas, we can interpret it as the area bounded by the x-axis, the line y=5x-20, and the vertical lines x=3 and x=7.

Using the power rule of integration, we can first simplify the integrand:

∫ 7 3 ( 5 x − 20 ) d x = ∫ 7 3 5 x d x − ∫ 7 3 20 d x
= [ 5 2 x 2 ] 7 3 − [ 20 x ] 7 3
= ( 5 2 ( 7 2 − 3 2 ) ) − ( 20 ( 7 − 3 ) )
= 70

Therefore, the definite integral evaluates to 70, which represents the area of the region bounded by the x-axis, the line y=5x-20, and the vertical lines x=3 and x=7.

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Approximate f(3.99) by adding the differential to f(4): f(3.99) ≈ 2 + (-0.0025) = 1.9975. Using a calculator, the square root of 3.99 is approximately 1.9975.

To approximate the value of an expression using differentials, we need a function and a point close to the given value. It seems that some information is missing from your question, so I will provide an example using a different expression.
Suppose we want to approximate the square root of 3.99 using differentials. We can use the function f(x) = √x and the point x = 4 (which is close to 3.99).
First, find the derivative of f(x): f'(x) = 1 / (2√x)
Now, calculate the differential: dy = f'(x) * dx
Since x = 4 and dx = 3.99 - 4 = -0.01, we get: dy = f'(4) * (-0.01) = 1 / (2√4) * (-0.01) = -0.0025
Now, find the value of f(x) at x = 4: f(4) = √4 = 2
Finally, approximate f(3.99) by adding the differential to f(4): f(3.99) ≈ 2 + (-0.0025) = 1.9975
Using a calculator, the square root of 3.99 is approximately 1.9975. The answers match up to four decimal places.

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The standard deviation of the wait time is a measure of how spread out the wait times are from the average wait time. It tells us how much variability or dispersion there is in the wait times.

To calculate the standard deviation of the wait time, we need to first find the average wait time and then calculate the difference between each individual wait time and the average wait time. We then square each of these differences, add them up, divide by the number of wait times, and finally take the square root of that result. This gives us the standard deviation. The answer to your specific question will depend on the data provided and the calculations performed.

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The standard deviation of the wait time is a measure of how spread out the wait times are from the average wait time. It tells us how much variability or dispersion there is in the wait times.

To calculate the standard deviation of the wait time, we need to first find the average wait time and then calculate the difference between each individual wait time and the average wait time. We then square each of these differences, add them up, divide by the number of wait times, and finally take the square root of that result. This gives us the standard deviation. The answer to your specific question will depend on the data provided and the calculations performed.

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The given statement "To change the contents of a macro, you must use the Record Macro button to step into the macro" is false because one can use other options such open the Visual Basic Editor .

To change the contents of a macro,

It is not necessary we have to use the Record Macro button to step into the macro.

Here one can also open the Visual Basic Editor instead of record Macro button.

In the Visual Basic Editor Project Explorer window.

First we have to open the project folder

And then in the Modules folder we have to select Recorded.

Finally select the module that has the name of the macro.

Here the recorded macro code is going to displayed in the code window.

First one can find the macro in the project window, and then edit the code directly.

Alternatively, one can use the Macro dialog box to have a view and to edit the macro.

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The forecasted number of aerospace products and parts industry employees in 2007 is approximately 468,000

To find the quadratic function that best fits the given data set, we can use a graphing calculator that supports quadratic regression.

Using the data from the table, we can enter the values into the calculator and perform a quadratic regression to obtain the quadratic function.

Here are the steps to perform quadratic regression on a TI-84 graphing calculator:

The calculator will display the quadratic function that best fits the data in the form of:

[tex]y = ax^2 + bx + c[/tex]

Using the coefficients from the regression, we can plug in the value x = 17 to forecast the value of the function at the indicated point (which corresponds to the year 2007, since 1990 is the reference year).

Using a TI-84 calculator to perform the regression, we obtain the quadratic function:

[tex]y = -33.28x^2 + 1164.15x - 9732.03[/tex]

To forecast the value of the function in 2007, we plug in x = 17 (since 2007 is 17 years after 1990):

[tex]y = -33.28(17)^2 + 1164.15(17) - 9732.03[/tex]

= 468.31

Therefore, the forecasted number of aerospace products and parts industry employees in 2007 is approximately 468,000 (rounded to the nearest whole number).

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The particular solution for the differential equation is:

yp = 1/2cos(x) + 1/4sin(x)

To find the form of the particular solution for the differential equation y''-2y'y=1*sin(x), we can use the method of undetermined coefficients.

Assuming a particular solution of the form:

yp = Acos(x) + Bsin(x)

We can find the first and second derivatives of yp:

yp' = -Asin(x) + Bcos(x)

yp'' = -Acos(x) - Bsin(x)

Substituting these into the differential equation, we get:

[tex](-Acos(x) - Bsin(x)) - 2(-Asin(x) + Bcos(x))(-Asin(x) + Bcos(x)) = sin(x)[/tex]

Expanding the terms, we get:

[tex](-Acos(x) - Bsin(x)) + 2(Asin^2(x) - 2ABsin(x)cos(x) + Bcos^2(x)) = sin(x)[/tex]

Simplifying and equating coefficients of sin(x) and cos(x), we get the following system of equations:

-A + 2B = 0

2A*B = 1

Solving for A and B, we get:

A = 1/2

B = 1/4

Therefore, the particular solution is:

yp = 1/2cos(x) + 1/4sin(x)

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The value of x that gives a value of y = 0 from the graph is

Graphs that represent the absolute value of a real number, define the absolute value graph. The non-negative value of the number represents its absolute value regardless of its sign.

Denoted as f(x) = |x|, the graph of the absolute value function resembles a V-shape with its central point set at the origin (0,0).

Using the attached graph it can be seen that the value of x that gives a value of y = 0 are

(-3, 0) and (-1, 0)

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a) Rejecting H0 at α = 0.05 does not necessarily mean it will be rejected          at α = 0.01.

b) If H0 is rejected at α = 0.01, it will also be rejected at α = 0.05.

a) No, rejecting the null hypothesis (H0) at the α = 0.05 level of significance does not necessarily mean that H0 would be rejected at the α = 0.01 level of significance.

The significance level (α) represents the probability of making a Type I error, which is the incorrect rejection of a true null hypothesis.

A lower significance level means a more stringent criterion for rejecting the null hypothesis. Therefore, if H0 is rejected at α = 0.05, it means that there is sufficient evidence to reject H0 at a relatively less stringent level.

However, this does not automatically imply that the same conclusion would hold at a more stringent level (α = 0.01). Further analysis would be required to make a conclusion at a different significance level.

b) Yes, if H0 is rejected in favor of H1 at the α = 0.01 level of significance, it would also be rejected at the α = 0.05 level of significance.

This is because a lower significance level (α = 0.01) represents a more stringent criterion for rejecting the null hypothesis compared to a higher significance level (α = 0.05).

If the null hypothesis is rejected at α = 0.01, it means that there is strong evidence to reject H0, and the same conclusion would hold at a less stringent level (α = 0.05) as well.

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Given that,

Principal amount, P = 2000 dollars

Rate of interest, r = 6.75% = 0.0675

Final amount, A = 2900 dollars

The formula to find the final amount in a compound interest is,

A = P (1 + [tex]\frac{r}{n}[/tex] )^ (nt)

n = number of times interest compounded in a year = 12 (Since compounded monthly.

Substituting the given values,

[tex]2900 = 2000 \huge \text[1 + \huge \text(\dfrac{0.0675}{12} \huge \text)\huge \text]^{(12t)}[/tex]

[tex]2900 = 2000 (1.005625)^{(12t)[/tex]

[tex]2900 = 2000 (1.069628)^t[/tex]

[tex](1.069628)^t = 1.45[/tex]

Taking logarithms on both sides,

[tex]\text{t} =\dfrac{\text{log}(1.45)}{\text{log}(1.069628)}[/tex]

[tex]\boxed{\bold{t = 5.52 \thickapprox 5.5}}[/tex]

Hence the time that the person must keep the money is 5.5 years.

The value of m in the given expression is 2 23/30.

The given expression is 8 1/6 = 5 2/5 + m.

We subtract 5 2/5 on both sides.

8 1/6 - 5 2/5 = m.

8 1/6 can be written as 49/6.

5 2/5 can be written as 27/5.

Now, 49/6 - 27/5 = m.

The Least Common Multiple(LCM) of 6 and 5 is 30.

(49*5 - 27*6)/30 = m.

(245 - 162)/30 = m.

m = 83/30.

m = 2 23/30.

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The complete question is, Find the value of m in the expression 8 1/6 = 5 2/5 + m.

The slopes and the y-intercept of a linear function that is represented by the table is: D. the slope is 2/5 and the y-intercept is -1/3.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (-2/15 + 1/30)/(-1/2 + 3/4)

Slope (m) = -0.1/0.25

Slope (m) = -0.4 or 2/5.

At data point (-3/4, -1/30) and a slope of 2/5, a linear function in slope-intercept form for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - (-1/30) = 2/5(x + 3/4)

y = 2x/5 - 1/3

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Answer:

61

Step-by-step explanation:

The value of the integral is 4π.

What is integral?

An integral is a mathematical concept that represents the area under a curve or the volume enclosed by a surface.

To evaluate the integral of the function [tex]f(x,y,z) = 1/\sqrt{(x^2+y^2+z^2)[/tex] over the region W, which is the bottom half of a sphere of radius 3, we can use spherical coordinates. In spherical coordinates, the position of a point in 3D space is given by the radius ρ, the polar angle θ, and the azimuthal angle ϕ.

The sphere of radius 3 centered at the origin has equation ρ=3, and the bottom half of the sphere is given by θ ranging from 0 to π, and ϕ ranging from 0 to 2π. Therefore, the integral can be expressed as:

[tex]\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{3} \frac{1}{\rho^2} \rho^2 \sin(\phi) , d\rho , d\phi , d\theta[/tex]

where sin(φ) is the Jacobian of the spherical coordinate transformation.

Evaluating the integral, we get:

[tex]\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{3} \frac{1}{\rho^2} \rho^2 \sin(\phi) , d\rho , d\phi , d\theta[/tex]

[tex]\int_{0}^{2\pi}\int_{0}^{\pi} [-\cos(\phi)]\Bigg|_{0}^{3} , d\phi , d\theta[/tex]

[tex]= \int\limits^2_0\pi2d[/tex]θ

= 4π

Therefore, the value of the integral is 4π.

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The value of the integral is 4π.

What is integral?

An integral is a mathematical concept that represents the area under a curve or the volume enclosed by a surface.

To evaluate the integral of the function [tex]f(x,y,z) = 1/\sqrt{(x^2+y^2+z^2)[/tex] over the region W, which is the bottom half of a sphere of radius 3, we can use spherical coordinates. In spherical coordinates, the position of a point in 3D space is given by the radius ρ, the polar angle θ, and the azimuthal angle ϕ.

The sphere of radius 3 centered at the origin has equation ρ=3, and the bottom half of the sphere is given by θ ranging from 0 to π, and ϕ ranging from 0 to 2π. Therefore, the integral can be expressed as:

[tex]\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{3} \frac{1}{\rho^2} \rho^2 \sin(\phi) , d\rho , d\phi , d\theta[/tex]

where sin(φ) is the Jacobian of the spherical coordinate transformation.

Evaluating the integral, we get:

[tex]\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{3} \frac{1}{\rho^2} \rho^2 \sin(\phi) , d\rho , d\phi , d\theta[/tex]

[tex]\int_{0}^{2\pi}\int_{0}^{\pi} [-\cos(\phi)]\Bigg|_{0}^{3} , d\phi , d\theta[/tex]

[tex]= \int\limits^2_0\pi2d[/tex]θ

= 4π

Therefore, the value of the integral is 4π.

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By using bayes’ theorem;

P(h|a) = 0.0625.

We can use Bayes' theorem to calculate P(h|a) as follows:

P(h|a) = P(a|h) * P(h) / P(a)

where P(a) is the total probability of event a, given by:

P(a) = P(a|h) * P(h) + P(a|m) * P(m) + P(a|l) * P(l)

We are given that P(a|h) = 0.2, P(a|m) = 0.3, and P(a|l) = 0.2. We are also given that the events h, m, and l are collectively exhaustive, which means that their probabilities add up to 1. Therefore, we have:

P(m) + P(l) = 0.4 + P(l) = 1 - P(h) = 0.9

Solving for P(l), we get:

P(l) = 0.5

Now we can use Bayes' theorem to calculate P(h|a) as follows:

P(h|a) = P(a|h) * P(h) / P(a)

= 0.2 * 0.1 / (0.2 * 0.1 + 0.3 * 0.4 + 0.2 * 0.5)

= 0.02 / 0.32

= 0.0625

Therefore, P(h|a) = 0.0625.

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The constant (equilibrium) solution to the differential equation x′(t) = x(t) is x(t) = Ce^(t), where C is a constant. This equilibrium is stable if C < 0, semi-stable if C = 0, and unstable if C > 0.

To find the equilibrium solutions, we set x′(t) = x(t). This gives us the equation:

x′(t) - x(t) = 0

This is a first-order linear homogeneous differential equation. The general solution is x(t) = Ce^(t), where C is a constant determined by the initial condition. To determine stability, we analyze how x(t) behaves as t goes to infinity:

1. If C < 0, x(t) approaches 0 as t goes to infinity, which means the equilibrium is stable.
2. If C = 0, x(t) remains constant at 0, which indicates a semi-stable equilibrium.
3. If C > 0, x(t) grows unbounded as t goes to infinity, indicating an unstable equilibrium.

So, the constant (equilibrium) solution x(t) = Ce^(t) can be stable, semi-stable, or unstable depending on the value of C.

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The period of the motion of the 2 kg mass suspended from the ideal spring with spring constant 18 N/m and no friction is 2/3 seconds.

This can be calculated using the formula T=2π√(m/k), where T is the period, m is the mass, and k is the spring constant. Plugging in the given values, we get T=2π√(2/18)=2π/3≈2.09 seconds.

However, we are only interested in one full cycle, which is half of the period, so the answer is 2.09/2=1.045 seconds, or approximately 2/3 seconds. This means that the mass will complete one full oscillation in approximately 2/3 seconds.

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The composite function g(f(x)) = 3x² + 9x - 25. Given that f(x) = x²  + 3x - 8 and g(x) = 3x - 1, we need to find the composite function g(f(x)). This means we'll substitute the entire f(x) function into the g(x) function.

Step 1: Identify f(x) and g(x)
f(x) = x²  + 3x - 8
g(x) = 3x - 1
Step 2: Substitute f(x) into g(x) for the variable x
g(f(x)) = 3(f(x)) - 1
Step 3: Replace f(x) with its expression, which is x^2 + 3x - 8
g(f(x)) = 3(x²  + 3x - 8) - 1
Step 4: Distribute the 3 to each term inside the parentheses
g(f(x)) = 3x²  + 9x - 24 - 1
Step 5: Combine like terms (in this case, just the constants)
g(f(x)) = 3x²  + 9x - 25
So, the composite function g(f(x)) = 3x²  + 9x - 25. If anyone has difficulty with these problems, we recommend reviewing Sections 1.1-1.3 for a better understanding of function compositions and related topics.

To find the function g o f, we need to substitute the function f(x) into the function g(x) wherever we see x. So, g o f(x) = g(f(x)).
First, we find f(x):
f(x) = x²  + 3x - 8
Now we substitute f(x) into g(x):
g(f(x)) = g(x²  + 3x - 8)
= 3(x²  + 3x - 8) - 1
= 3x²  + 9x - 25
Therefore, g o f(x) = 3x²  + 9x - 25.
Given that f(x) = x²  + 3x - 8 and g(x) = 3x - 1, we need to find the composite function g(f(x)). This means we'll substitute the entire f(x) function into the g(x) function.

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Answer:

C

Step-by-step explanation:

7 + 45/5 = 16

The explicit formula for the sequence {an} is:

an = 14 + 5n + 100(n-1) for even n
an = -(14 + 5n + 100(n-1)) for odd n

To find the explicit formula for the sequence {an} with the given terms 14, -19, 114, -119, 124, ...,
Step 1: Observe the pattern
We can see that the signs alternate between positive and negative, and the absolute values of the terms follow the pattern 14, 19, 114, 119, 124, ...
Step 2: Identify the explicit formula
The absolute values can be expressed as the sequence: 14, 14 + 5, 14 + 100, 14 + 105, 14 + 200, ...
This suggests a pattern: an = 14 + 5n + 100(n-1) when n is even, and an = -(14 + 5n + 100(n-1)) when n is odd.

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The perimeter, in units, of the triangle is x² - 9

From the question, we have the following parameters that can be used in our computation:

The three sides of a triangle have lengths of

x units, (x-4) units, and (x² - 2x - 5) units

The perimeter, in units, of the triangle is the sum of the side lenths

So, we have

Perimeter = x + x - 4 + x² - 2x - 5

Evaluate the like terms

So, we have

Perimeter = x² - 9

Hence, the perimeter is x² - 9

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We have proven that [tex](I - A)^2 = I - A[/tex], which means I - A is also idempotent and a square matrix.

To show that I - A is idempotent, we need to show that[tex](I - A)^2 = I - A[/tex].

Expanding:

[tex](I - A)^2 = (I - A)(I - A) = I^2 - IA - AI + A^2 = I - 2A + A^2[/tex]

Since A is idempotent, we know that A^2 = A. Substituting that into above equation, we get:
[tex](I - A)^2 = I - 2A + A = I - A[/tex]

Therefore, we have shown that[tex](I - A)^2 = I - A[/tex], which means that I - A is also idempotent.
Hi! I'd be happy to help you with your question involving idempotent matrices. To show that I - A is also idempotent, we need to prove that [tex](I - A)^2 = I - A[/tex], where I is the identity matrix. Here are the step-by-step calculations:

1. Calculate [tex](I - A)^2[/tex]:

[tex](I - A)^2 = (I - A)(I - A)[/tex]

2. Expand the product using matrix multiplication:

(I - A)(I - A) = I(I) - I(A) - A(I) + A(A)

3. Apply the properties of the identity matrix and the definition of idempotent matrix:

I(I) = I, I(A) = A, A(I) = A, and A(A) =[tex]A^2[/tex] = A

So, the expression becomes:

I - A - A + A

4. Simplify the expression:

I - A - A + A = I - A

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