(a) The integers (aeij) = 0 for j ≠ i, demonstrating that AE is the matrix whose jth column equals the ith column of A and all other columns are zero.
To prove that EijA is the matrix whose ith row equals the jth row of A and all other rows are zero, we can consider the matrix multiplication between Eij and A.
Let's denote the elements of A as A = [aij] and the elements of Eij as Eij = [eijk]. The matrix product EijA can be calculated as follows:
(EijA)ij = ∑k eijk * akj
Since Eij has a 1 in row i and column j, and zeros elsewhere, only the term with k = j contributes to the sum. Thus, the above expression simplifies to:
(EijA)ij = eiji * ajj = 1 * ajj = ajj
For all other rows, since Eij has zeros, the sum evaluates to zero. Therefore, (EijA)ij = 0 for i ≠ j.
This shows that EijA is the matrix whose ith row equals the jth row of A and all other rows are zero.
Similarly, to prove that AE is the matrix whose jth column equals the ith column of A and all other columns are zero, we can perform matrix multiplication between A and E.
Let's denote the elements of AE as AE = [aeij]. The matrix product AE can be calculated as:
(aeij) = ∑k aik * ekj
Again, since E has a 1 in row j and column i, only the term with k = i contributes to the sum. Thus, the expression simplifies to:
(aeij) = aij * eji = aij * 1 = aij
For all other columns, since E has zeros, the sum evaluates to zero.
(b) I contains the identity matrix, which means that I is equal to M₁(F).
Since A was an arbitrary nonzero matrix, this implies that every nonzero matrix generates the entire space M₁(F). Hence, Mn(F) is a simple ring, meaning it has no nontrivial ideals.
Let A ∈ M₁(F) be a nonzero matrix, and let I be the ideal generated by A.
We need to show that Eij ∈ I for each integer i in the range 1 ≤ i ≤ n.
Consider the product AEij. As shown in part (a), AEij is the matrix whose jth column equals the ith column of A and all other columns are zero. Since A is nonzero, the jth column of A is nonzero as well. Therefore, AEij is nonzero, implying that AEij ∉ I.
Since AEij ∉ I, it follows that Eij ∈ I for each i in the range 1 ≤ i ≤ n.
Now, we know that Eij ∈ I for all i in the range 1 ≤ i ≤ n. This means that I contains all matrices with a single nonzero entry in each row.
Consider the identity matrix In. Each entry in the identity matrix can be obtained as a sum of matrices from I. Specifically, each entry (i, i) in the identity matrix can be obtained as the sum of Eii matrices, which are all in I.
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HELP PLEASEEEE IT WOULD HELP ME OUT A LOT
Answer:
Step-by-step explanation:
a company staff consists of 20 accountants, 12 economists and 4 secretaries. a staff is chosen at random. Find the probability that the staff is an accountant. With solution.
Answer:
5/9
Step-by-step explanation:
Number of accountants = 20
Number of economists = 12
Number of secretaries = 4
Total number of Staffs = 20 + 12 + 4 = 36 staffs
Probability = required outcome / Total possible outcomes
Required outcome = number of accountants
Total possible outcomes = total number of staffs
P(selecting an economist) = 20 / 36 = 5 / 9
The probability that the staff is an accountant is 5/9.
Given
A company staff consists of 20 accountants, 12 economists and 4 secretaries. a staff is chosen at random.
Probability;Probability is defined as the number of observations and total number of observation.
Total number of Staffs = 20 + 12 + 4 = 36 staffs.
The following formula is used to determine the probability;
[tex]\rm Probability=\dfrac{Accountant \ staff}{Total \ number \ of \ staff}[/tex]
Substitute all the values in the formula;
[tex]\rm Probability=\dfrac{Accountant \ staff}{Total \ number \ of \ staff}\\\\\rm Probability=\dfrac{20}{36}\\\\\rm Probability=\dfrac{5}{9}[/tex]
Hence, the probability that the staff is an accountant is 5/9.
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the functions y=x^2+ (c/x^2) are all solutions of equation: xy′ 2y=4x^2, (x>0). find the constant c which produces a solution which also satisfies the initial condition y(6)=4. c= ______--
The constant c that produces a solution satisfying the initial condition y(6) = 4 is c = 24.
To find the constant c that satisfies the given equation and the initial condition, we need to substitute the function y = x² + (c/x²) into the differential equation and solve for c. The given equation is xy' * 2y = 4x².
First, we differentiate y with respect to x to find y',
y = x² + (c/x²)
y' = 2x - (2c/x³)
Now we substitute y and y' into the differential equation,
xy' * 2y = 4x²
x(2x - (2c/x³)) * 2(x² + (c/x²)) = 4x²
Simplifying,
2x³ - 2cx + 4c = 4x²
Rearranging,
2x³ - 4x² - 2cx + 4c = 0
Now we substitute x = 6 and y = 4 (from the initial condition y(6) = 4) into the equation,
2(6)³ - 4(6)² - 2(6)c + 4c = 0
432 - 144 - 12c + 4c = 0
288 - 8c = 0
8c = 288
c = 36
Therefore, the constant c that produces a solution satisfying the initial condition y(6) = 4 is c = 36.
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Given: ΔWXY is isosceles with legs WX and WY; ΔWVZ is isosceles with legs WV and WZ. Prove: ΔWXY ~ ΔWVZ. Complete the steps of the proof.
a. ASA (Angle-Side-Angle)
b. SAS (Side-Angle-Side)
c. SSS (Side-Side-Side)
d. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
We have proven that ΔWXY is similar to ΔWVZ using the ASA criterion.
We have,
To prove that ΔWXY is similar to ΔWVZ, we can use the ASA (Angle-Side-Angle) criterion.
Here are the steps of the proof:
Proof:
- Given: ΔWXY is isosceles with legs WX and WY; ΔWVZ is isosceles with legs WV and WZ.
Since ΔWXY is isosceles, we have WX ≅ WY. (Given)
Since ΔWVZ is isosceles, we have WV ≅ WZ. (Given)
We also know that ΔWXY and ΔWVZ share the common side segment WZ. (Common side)
Let's consider the angles: ∠WXY and ∠WVZ. Since ΔWXY is isosceles, we have ∠WXY ≅ ∠WYX. (Isosceles triangle property)
Similarly, since ΔWVZ is isosceles, we have ∠WVZ ≅ ∠WZV. (Isosceles triangle property)
Now, we have two pairs of congruent angles: ∠WXY ≅ ∠WYX and ∠WVZ ≅ ∠WZV.
We already know that WX ≅ WY and WV ≅ WZ.
By the ASA criterion, if two pairs of corresponding angles and the included side are congruent, then the triangles are similar.
Applying the ASA criterion, we conclude that ΔWXY ~ ΔWVZ. (Angle-Side-Angle)
Therefore,
We have proven that ΔWXY is similar to ΔWVZ using the ASA criterion.
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Write an equivalent expression for 5+2+2x+2
please help me
Answer:
2x+9
Step-by-step explanation:
You have to combine like terms. 2x stays the same because there are no other like terms, but 5, 2, and 2 can be added together to make 9
Solve the equation.
[tex] {3}^{4(m + 1)} + {3}^{4m} - 246 = 0 \\ [/tex]
[tex]3^{4m+4}+3^{4m}=246\\ (3^{4}+1)*3^{4m}=246\\82*3^{4m}=246\\3^{4m}=3\\m=\frac{1}{4}[/tex]
Dani has only 1/2 of a cup of baking mix. She needs 3/4 of a cup for one batch of muffins. Which solution method shows how much of a batch of muffins Dani can make?
Answer:
Step-by-step explanation:
Dani has only 1/2 of a cup of baking mix. She needs 3/4 of a cup for one batch of muffins. Which solution method shows how much of a batch of muffins Dani can make?
Please help me, I’m struggling.
A: 116 square cm
B: 106 square cm
C: 143 square cm
Answer:
The answer is A. 116 square cm
Step-by-step explanation:
Consider the following quadratic models: (1) y = 1 – 2x + x2 (2) y = 1 + 2x + x2 (3) y = 1 + x2 (4) y = 1 - 42 (5) y = 1 + 372 y a. Graph each of the quadratic models, side by side, on the same sheet of graph paper
All the graph of equations are shown in figure.
We have to given that,
All the quadratic equations are,
1) y = 1 - 2x + x²
2) y = 1 + 2x + x²
3) y = 1 + x²
4) y = 1 - x²
5) y = 1 + 3x²
We can see that,
All the equation form a quadratic equation.
Hence, Each graph shows a parabola.
Therefore, All the graph of equations,
1) y = 1 - 2x + x²
2) y = 1 + 2x + x²
3) y = 1 + x²
4) y = 1 - x²
5) y = 1 + 3x²
are shown in figure.
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Calculate the mean, median, and range of the data in the dot plot.
Answer:
median = 5
range = 2
mean = 11
Step-by-step explanation:
Adjust the equation so the line passes through the points.
pleaae help explain and write clearly thank you
you need to write a post describing either the column space or the null space of a matrix.
The null space of a matrix is the set of all vectors that satisfy the equation Ax = 0.
The null space of a matrix is the set of all vectors that satisfy the equation Ax = 0. In other words, the null space of a matrix A is the set of all solutions x to the equation Ax = 0. The null space of a matrix is also known as the kernel of a matrix. It is a subspace of the vector space R^n. The null space of a matrix can be used to determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions. If the null space of a matrix is the zero vector, then the system has a unique solution. If the null space of a matrix is non-empty, then the system has infinitely many solutions. A matrix is an array of numbers that has been set up in rows and columns to make a rectangular shape. The elements, or entries, of the matrix are the integers. In addition to numerous mathematical disciplines, matrices find extensive use in the fields of engineering, physics, economics, and statistics. In computer graphics, where they have been used to describe picture rotations and other transformations, matrices have vital applications as well.
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(please help)!!!!!!!!!!!!!!
Answer
use trigonometry to find both the answers.
i have done the working out in the picture hope it helps...
Step-by-step explanation:
10. Use the two given poInts and calculate the slope.
(7,2), (6,1)
Answer:
The answer is m=1 , the slope is 1
Step-by-step explanation:
9)
Consider
-196/14
Which THREE statements are correct?
A)
The quotient is 14.
B)
The quotient is -14.
The quotient is -
D)
- 196
is equivalent to the expression.
14
E)
196
-14
is equivalent to the expression.
Fill in the blanks:- If y = 2 - x + x2 + 8ex is a solution of a homogeneous fourth-order linear differential equation with constant coefficients, then the roots of the auxiliary equation are_________ .
The roots of the auxiliary equation for a homogeneous fourth-order linear differential equation with constant coefficients, given that the solution is y = 2 - x + x^2 + 8e^x, are -1, -1, -2, and -2.
For a homogeneous linear differential equation with constant coefficients, the auxiliary equation is obtained by replacing the derivatives of y with powers of the variable. In this case, since the given solution is y = 2 - x + x^2 + 8e^x, we differentiate y with respect to x to obtain the derivatives.
The fourth-order linear differential equation corresponds to the fourth power of the variable, which is x. Therefore, the auxiliary equation is a polynomial equation of degree four. To find the roots of the auxiliary equation, we set the polynomial equal to zero and solve for x.
The roots of the auxiliary equation for this particular solution, after solving the polynomial equation, are -1, -1, -2, and -2. These values represent the roots of the characteristic equation and are crucial in determining the form of the general solution of the differential equation.
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Let z = (a + ai)(b + b/3i) where a and b are positive real numbers. Without using a calculator, determine arg z.
The argument (arg) of the complex number z = (a + ai)(b + b/3i), where a and b are positive real numbers, is π/6 radians or 30 degrees.
To determine the argument (arg) of the complex number z = (a + ai)(b + b/3i), where a and b are positive real numbers, we can simplify the expression and find the argument without using a calculator.
First, expand the product (a + ai)(b + b/3i):
z = (a + ai)(b + b/3i)
= ab + ab/3i + abi - ab/3
Combining like terms, we get:
z = (ab - ab/3) + (ab/3 + ab)i
= (2ab/3) + (ab/3)i
Now, we have the complex number z in the form z = x + yi, where x = 2ab/3 and y = ab/3.
To compute the argument (arg) of z, we can use the definition of the argument as the angle θ between the positive real axis and the line connecting the origin to the complex number z in the complex plane.
Since a and b are positive real numbers, both x and y are positive.
The argument (arg) of z can be determined as:
arg z = arctan(y/x)
= arctan((ab/3) / (2ab/3))
= arctan(1/2)
= π/6
Therefore, without using a calculator, the argument (arg) of the complex number z = (a + ai)(b + b/3i) is π/6 radians or 30 degrees.
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Describe a situation that can be represented by the integer - 6
Answer:
One Chilly Night, Elijah Was Sleeping but Then Woke Up. He Realized It Was 45 Degrees According To the AC Monitor, So He Changed it To 70 Degrees But The Monitor Broke And Changed to -6 DEGREES!!
Step-by-step explanation:
A situation that can be represented by the integer - 6 was envisaged.
What are integers?An integer is a whole number (not a fractional number) that can be positive, negative, or zero.
One day maths teacher decided to take a random test
There were 16 questions each of 4 marks and 1 mark deduction for each wrong answer, with no penalties for non-attempt.
David was so good at guessing. He guessed all the 16 answers out of which 2 were correct and 14 were incorrect.
So, the total marks that David got = 4(2)-1(14) = 8-14 =-6
Thus, a situation that can be represented by the integer - 6 was envisaged.
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Solve the system using substitution. Show all work.
( 4х + 5y = 7
у = 3х + 9
Answer:
(4x+5y=7
y=3x+9
4x+5(3x+9)=7
4x+15x+45=7
19x=7-45
19x= -38
19×/19=-38/19
x= -2
whlie y=3x+9
y=3(-2)+9
y= -6+9
y=3 end solution
x= -2,y= 3
Write a linear function f with given values. F(3)=-4, f(5)= -4
Answer:
y = -4
Step-by-step explanation:
dy/dx gives you slope
(-4)-(-4)/5-3 = 0/2 ----> slope = 0
y = mx+b
m = 0
y = 0x+b
y = b
as it says F(3) and F(5) = -4 b must be -4
so you end up with y = -4
Lacey opened a savings account and deposited $100.00. The
account earns 6% interest, compounded monthly. If she wants
to use the money to buy a new bicycle in 2 years, how much
will she be able to spend on the bike?
Round your answer to the nearest cent.
Answer:
$244.00
Step-by-step explanation:
- gave bank $100
- +6% of $100 per month
- 6% of $100 = $6 (earns $6 per month)
- twelve months in a year, so 12 x $6 = $72 per year
- $100 + $72 + $72 = $244
If Lacey wants to utilize her money to purchase a new bicycle in two years, she will spend $244 on the bike.
"6 less than the quotient of a number and 5"
Answer:
[tex]\frac{n}{5} - 6[/tex]
Step-by-step explanation:
Quotient of a number and 5 means n/5
6 less than n/5 means n/5 - 6
Answer:
the product of a triple a number and 19
Step-by-step explanation:
but i'm not 100% sure so don't quote me
Assume the population is normally distributed. Given a sample size of 225, with a sample mean of 750 and a standard deviation of 30, we perform the following hypothesis test.
H0: μ = 745
Ha: μ ≠ 745
a) Is this test for the population proportion, mean, or standard deviation? What distribution should you apply for the critical value?
b) What is the test statistic?
c) What is the p-value?
d) What is your conclusion of the test at the α = 0.1005 level? Why?
We need to determine whether the test is for the population proportion, mean, or standard deviation, and what distribution should be applied for the critical value.
a) This test is for the population mean since we are comparing the sample mean to a hypothesized population mean. To find the critical value, we apply the t-distribution since the population standard deviation is un known, and we are working with a sample.
b) The test statistic for comparing means is calculated using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / √sample size).
Substituting the given values, we have:
t = (750 - 745) / (30 / √225) = 5 / 2 = 2.5.
c) To find the p-value, we compare the absolute value of the test statistic to the critical value associated with the significance level. Since the significance level α is not specified, we cannot directly calculate the p-value without knowing the critical value or α.
d) Without the critical value or the specific significance level, we cannot determine the conclusion of the test. The conclusion is drawn by comparing the p-value to the significance level α. If the p-value is less than α, we reject the null hypothesis, and if the p-value is greater than α, we fail to reject the null hypothesis.
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Write the difference of 8 and 2 times m
Answer:
Your answer would be A
Step-by-step explanation:
Draw the graph of y = log, (z) - 4
The graph of y = log(z) - 4 is a downward shifted logarithmic function with a vertical asymptote at z = 0.
The equation y = log(z) - 4 represents a logarithmic function. The graph of a logarithmic function typically consists of a vertical asymptote, which is a vertical line that the graph approaches but never crosses. In this case, the vertical asymptote occurs at z = 0, as the logarithm of a negative number is undefined.
The graph is vertically shifted downward by 4 units, which means that the entire graph is shifted downward by 4 units compared to the standard logarithmic function. This shift moves the graph downward parallel to the y-axis.
The domain of the function is the set of positive real numbers (z > 0), as the logarithm is defined only for positive values. The range of the function is all real numbers, as the graph extends infinitely in both the positive and negative y-directions.
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please tell me where to plot the points and what the solution will be.
solve the following question
Answer:
g) [tex]u^{4}\cdot v^{-1}\cdot z^{3}[/tex], h) [tex]\frac{(x+4)\cdot (x+2)}{3\cdot (x-5)}[/tex]
Step-by-step explanation:
We proceed to solve each equation by algebraic means:
g) [tex]\frac{u^{5}\cdot v}{z}\div \frac{u\cdot v^{2}}{z^{4}}[/tex]
1) [tex]\frac{u^{5}\cdot v}{z}\div \frac{u\cdot v^{2}}{z^{4}}[/tex] Given
2) [tex]\frac{\frac{u^{5}\cdot v}{z} }{\frac{u\cdot v^{2}}{z^{4}} }[/tex] Definition of division
3) [tex]\frac{u^{5}\cdot v\cdot z^{4}}{u\cdot v^{2}\cdot z}[/tex] [tex]\frac{\frac{a}{b} }{\frac{c}{d} } = \frac{a\cdot d}{b\cdot c}[/tex]
4) [tex]\left(\frac{u^{5}}{u} \right)\cdot \left(\frac{v}{v^{2}} \right)\cdot \left(\frac{z^{4}}{z} \right)[/tex] Associative property
5) [tex]u^{4}\cdot v^{-1}\cdot z^{3}[/tex] [tex]\frac{a^{m}}{a^{n}} = a^{m-n}[/tex]/Result
h) [tex]\frac{x^{2}-16}{x^{2}-10\cdot x + 25} \div \frac{3\cdot x - 12}{x^{2}-3\cdot x -10}[/tex]
1) [tex]\frac{x^{2}-16}{x^{2}-10\cdot x + 25} \div \frac{3\cdot x - 12}{x^{2}-3\cdot x -10}[/tex] Given
2) [tex]\frac{\frac{x^{2}-16}{x^{2}-10\cdot x+25} }{\frac{3\cdot x - 12}{x^{2}-3\cdot x - 10} }[/tex] Definition of division
3) [tex]\frac{(x^{2}-16)\cdot (x^{2}-3\cdot x -10)}{(x^{2}-10\cdot x + 25)\cdot (3\cdot x - 12)}[/tex] [tex]\frac{\frac{a}{b} }{\frac{c}{d} } = \frac{a\cdot d}{b\cdot c}[/tex]
4) [tex]\frac{(x+4)\cdot (x-4)\cdot (x-5)\cdot (x+2)}{3\cdot (x-5)^{2}\cdot (x-4) }[/tex] Factorization/Distributive property
5) [tex]\left(\frac{1}{3} \right)\cdot (x+4)\cdot (x+2)\cdot \left(\frac{x-4}{x-4} \right)\cdot \left[\frac{x-5}{(x-5)^{2}} \right][/tex] Modulative and commutative properties/Associative property
6) [tex]\frac{(x+4)\cdot (x+2)}{3\cdot (x-5)}[/tex] [tex]\frac{a^{m}}{a^{n}} = a^{m-n}[/tex]/[tex]\frac{a}{b}\times \frac{c}{d} = \frac{a\cdot c}{b\cdot d}[/tex]/Definition of division/Result
If a graphed line passes through ordered pair
points (-6, 2) and (5, 4), what is the slope of
the line?
Answer:
I think it would be 2/11 . hope it helps u ^.^
What is the midpoint of line segment QU given Q(6, 3) and P(-6, -1).
Answer: Use cylindrical coordinates. Evaluate z dv, where E is enclosed by ... A: Solution:Given∫c3y sinx dx+5xdyFormula:∫cPdx+Qdy=∬ ∂Q∂x-∂P∂y dA ... z) if the midpoint of the line segment joining the two points (x, y, z) and (-6, -5, -4). ... A: A cone is a 3-D shape with a circular base and tapers smoothly over to an
Step-by-step explanation:
(5)
A plane contains the points A(1, 2, 8), B(-2, 3, 6), and C(5,-1, 4).
(a) Determine 2 vectors parallel to the plane.
(b) Determine 2 vectors perpendicular to the plane.
(C) Write a vector equation of the plane.
(d) Write a scalar equation of the plane.
(e) Determine if the point D(-7, 4, 0) is contained in the plane.
(f) Write an equation of the line through the y and z intercepts of the plane.
Answer:
7
Step-by-step explanation: