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Explanation:
Definition
The determinant of a square matrix is a single number that is computed (recursively) as the sum of products of the elements of a row or column and the determinants of their cofactors. The determinant of a single element is the value of that element.
The cofactor of an element in an n by n matrix is the (n-1) by (n-1) matrix that results when the row and column of that element are deleted. The "appropriate sign" of the element is applied to the cofactor matrix. The "appropriate sign" of an element is positive if the sum of its row and column numbers is even, negative otherwise. (Rows and columns are considered to be numbered 1 to n in an n by n matrix.)
Uses
The inverse of a square matrix is the transpose of the cofactor matrix, divided by the determinant. Hence if the determinant is zero, the inverse matrix is undefined. This means any system of equations the matrix might represent will have no distinct solution. (There may be zero solutions, or there may be an infinite number of solutions. The determinant by itself cannot tell you which.)
Cramer's Rule for the solution of linear systems of equations specifies that the value of any given variable is the ratio of the determinants of two matrices. The numerator matrix is the original matrix with the coefficients of the variable replaced by the constants in the standard-form equations; the denominator matrix is the original coefficient matrix. This rule lets you solve a system of 3 equations in 3 variables by computing 3+1 = 4 determinants, for example.
Let's look at an example.
If we wanted to solve this system of equations
[tex]\begin{cases}2x-y = 2\\x+y = 7\end{cases}[/tex]
Then it's equivalent to solving this matrix equation
[tex]\begin{bmatrix}2 & -1\\1 & 1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\7\end{bmatrix}[/tex]
We can then further condense that into the form
[tex]Aw = B[/tex]
Where,
[tex]A = \begin{bmatrix}2 & -1\\1 & 1\end{bmatrix}\\\\w = \begin{bmatrix}x\\y\end{bmatrix}\\\\B = \begin{bmatrix}2\\7\end{bmatrix}[/tex]
------------------------------------------
To solve the matrix equation Aw = B, we could compute the inverse matrix [tex]A^{-1}[/tex] and left-multiply both sides by this to isolate w.
So we'd go from [tex]Aw=B[/tex] to [tex]w = A^{-1}*B[/tex]. The order of multiplication is important.
For any 2x2 matrix of the form
[tex]P = \begin{bmatrix}a & b\\c & d\end{bmatrix}[/tex]
its inverse is
[tex]P^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}[/tex]
Notice the expression ad-bc in the denominator of that fractional term outside. This [tex]ad-bc[/tex] expression represents the determinant of matrix P. Some books may use the notation "det" to mean "determinant"
[tex]P^{-1} = \frac{1}{\det(P)}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}[/tex]
or you may see it written as
[tex]P^{-1} = \frac{1}{|P|}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}[/tex]
Those aren't absolute value bars, even if they may look like it.
Based on that, we can see that the determinant must be nonzero in order to compute the inverse of the matrix. Consequently, the determinant must be nonzero in order for Aw = B to have one solution.
If the determinant is 0, then we have two possibilities:
There are infinitely many solutions (aka the system is dependent)There are no solutions (the system is inconsistent)So a zero determinant would have to be investigated further as to which outcome would occur.
------------------------------------------
Let's return to the example and compute the inverse (if possible).
[tex]A = \begin{bmatrix}2 & -1\\1 & 1\end{bmatrix}\\\\A^{-1} = \frac{1}{2*1 - (-1)*1}\begin{bmatrix}1 & 1\\-1 & 2\end{bmatrix}\\\\A^{-1} = \frac{1}{3}\begin{bmatrix}1 & 1\\-1 & 2\end{bmatrix}\\\\[/tex]
In this case, the inverse does exist.
This further leads to
[tex]w = A^{-1}*B\\\\w = \frac{1}{3}\begin{bmatrix}1 & 1\\-1 & 2\end{bmatrix}*\begin{bmatrix}2\\7\end{bmatrix}\\\\w = \frac{1}{3}\begin{bmatrix}1*2+1*7\\-1*2+2*7\end{bmatrix}\\\\w = \frac{1}{3}\begin{bmatrix}9\\12\end{bmatrix}\\\\w = \begin{bmatrix}(1/3)*9\\(1/3)*12\end{bmatrix}\\\\w = \begin{bmatrix}3\\4\end{bmatrix}\\\\\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}3\\4\end{bmatrix}\\\\[/tex]
This shows that the solution is (x,y) = (3,4).
As the other person pointed out, you could use Cramer's Rule to solve this system. Cramer's Rule will involve using determinants and you'll be dividing over determinants. So this is another reason why we cannot have a zero determinant.
If f:X is 3x + b and ff(2) = 12, find the value of b
Answer:
[tex]b =6[/tex]
Step-by-step explanation:
Given
[tex]f(x) =3x + b[/tex]
[tex]f(2) = 12[/tex]
Required
Find b
[tex]f(2) = 12[/tex] implies that:
[tex]12 = 3 * 2 + b[/tex]
[tex]12 = 6 + b[/tex]
Collect like terms
[tex]b = 12 - 6[/tex]
[tex]b =6[/tex]
Determine the product of (46.2 × 10^-1) ⋅ (5.7 × 10^–6). Write your answer in scientific notation.
A)
2633.4 × 10^–5
B)
2.6334 × 10^–7
C)
2.6334 × 10^–1
D)
2.6334 × 10^–5
Step-by-step explanation:
here's the answer to your question
Does the point (0, 0) satisfy the equation y = 9x?
Answer:
yes it does
Step-by-step explanation:
because the equation y=9x does not have a y-intercept (all slopes come in the form y=mx+b -- it can be written differently though) and since there is no 'b' that means the y-intercept is 0. So whenever there is no y-intercept, the slope starts at 0.
how to graph quadratic relationship for h(x)=(x-1)^2-9
Answer:
use the formula y = a(x-h)^2 + k
the a stretches or flattens the parabola,
The h shifts left to right , and the k shifts up/down
Step-by-step explanation:
The A&M Hobby Shop carries a line of radio-controlled model racing cars. Demand for the cars is assumed to be constant at a rate of 60 cars per month. The cars cost $70 each, and ordering costs are approximately $15 per order, regardless of the order size. The annual holding cost rate is 20%.
Required:
a. Determine the economic order quantity and total annual cost under the assumption that no backorders are permitted.
b. Using a $45 per-unit per-year backorder cost, determine the minimum cost inventory policy and total annual cost for the model racing cars.
c. What is the maximum number of days a customer would have to wait for a backorder under the policy in part (b)? Assume that the Hobby Shop is open for business 300 days per year.
d. Would you recommend a no-backorder or a backorder inventory policy for this product? Explain.
Answer:
Step-by-step explanation:
A) Demand per month= 40 cars
Annual Demand (D)= 12*40 = 480
Fixed Cost per order (K)= 15
Holding Cost= 20% of cost= 60 *0.2 = 12
a. Economic Order Quantity=
Q^{*}={\sqrt {{\frac {2DK}{h}}}}
= √(2*480*15)/12
=34.64 ~ 35
Total Cost =P*D+K(D/EOQ)+h(EOQ/2) P= Cost per unit
= 60*480+ 15(480/35) + 12(35/2)
= 28800+ 205.71+ 210
=$29215.71
B). Backorder Cost (b)= $45
Qbo= Q* × √( b+h/ h)
= 35*√(12+45/ 45)
= 35* 1.12
=39.28 ~ 39
Shortage (S)= Qbo * (K/K+b)
= 39* (15/15+45)
= 39* 0.25
= 9.75
Total Cost Minimum=( bS2/ 2Qbo) + P (Qbo- S)2/2Qbo + K(D/Qbo)
=45* 9.752 / 2* 392 + 60 (39-9.75)2/ 2* 392 + 15 ( 480/39)
= 1.40+ 21.9.+ 184.61
=$207.91
C)Length of backorder days (d) = Demand ÷ amount of working days
d = 480 ÷ 300
d = 1.6
Calculate the backorders as the maximum number of backorders divided by the demand per day
s/d = 9.75/1.6 = 6.09 days (answer)
D) Calculate the difference in total between not using backorder:
$207.85 + $207.85 - 207.91 = $207.79
The saving in using backorder is $207.79.
Therefore I would recommend using a backorder
Find the length of AC on this triangle
Answer:
A
Step-by-step explanation:
Using the tangent ratio in the right triangle
tan12° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{AC}{BC}[/tex] = [tex]\frac{AC}{44}[/tex] ( multiply both sides by 44 )
44 × tan12° = AC , then
AC ≈ 9.35 ( to 2 dec. places )
Suppose Event A is taking 15 or more minutes to get to work tomorrow and Event B is taking less than 15 minutes to get to work tomorrow. Events A and B are said to be complementary events.
a. True
b. False
Answer:
Hence the answer is TRUE.
Step-by-step explanation:
If event A is taking 15 or more minutes to urge to figure tomorrow and event B is taking but a quarter-hour to urge to figure tomorrow, then events A and B must be complimentary events. this is often because the occurring of 1 is going to be precisely the opposite of the occurring of the opposite event and that they cannot occur simultaneously. In other words, events A and B are mutually exclusive and exhaustive.
Mathematically,
P(A) + P(B) = 1.
(a) 4x + 3y + 2x + 7y
Answer:
6x + 10y
Step-by-step explanation:
4x + 3y + 2x + 7y
=> (4x + 2x) + (3y + 7y)
=> 6x + 10y
At Dorcas's Hair Salon there are three hair stylists. 27% of the hair cuts are done by Martin, 30% are done by Jennifer, and 43% are done by Dorcas. Martin finds that when he does hair cuts, 6% of the customers are not satisfied. Jennifer finds that when she does hair cuts, 7% of the customers are not satisfied. Dorcas finds that when she does hair cuts, 3% of the customers are not satisfied. Suppose that a customer leaving the salon is selected at random. If the customer is not satisfied, what is the probability that their hair was done by Dorcas
Answer:
Dorcas's Hair Salon
If the customer is not satisfied, the probability that their hair was done by Dorcas is:
= 18.75%
Step-by-step explanation:
Number of hair stylists = 3
Martin Jennifer Dorcas Total
Percentage of haircuts
done 27% 30% 43% 100%
Percentage of dissatisfied
customers 6% 7% 3%
Proportion of dissatisfied
customers 37.5% (6/16) 43.75% (7/16) 18.75% (3/16)
If the customer is not satisfied, the probability that their hair was done by Dorcas
= 18.75%
What is the vertex of the graph of this function
y= -(x+2) (x+4)
Answer:
y=-(x+2)(x+4)
y=-(x^2+4x+2x+8)
y=-(x^2+6x+8)
y=-(x^2+4x+2x+8)
Y=-x(x+4)+2(x+4)
y=-(x+2)(X+4)
So vertex is (2,4)
I need help guys thanks so much
I think its A) (f+g)(z)=|2x+4|-2
Step-by-step explanation:
Identify the decimals labeled with letters A B and C on the scale
Answer:
A. 37.39 B. 37.41 C. 37.27
NO LINKS OR ANSWERING WHAT YOU DON'T KNOW. THIS IS NOT A TEST OR AN ASSESSMENT!!!. Please help me with these math questions. Chapter 10 part 2
3. How do solving for solving to a rational function differ from solving for solutions to a rational inequality? How they are similar?
4. How is the difference quotient of a function determined? And how is the difference quotient related to the secant line? Is there a pattern for the difference quotient of linear functions?
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Answer:
3. sign changes in the denominator need to be taken into account
4. difference quotient: (f(x+h) -f(x))/h; It is the slope of the secant line. For linear functions, the slope is constant, as is the difference quotient.
Step-by-step explanation:
3. When solving the equation f(x) = 0, where f(x) is a rational function, only the numerator zeros need to be considered.
When solving the inequality f(x) ≤ 0, or f(x) < 0, both numerator and denominator zeros need to be considered. As with solving any inequality, multiplying or dividing by a negative number changes the sense of the comparison.
Example
f(x) = x/(x-2) changes sign at both x=0 and x=2. Then three regions need to be considered when solving f(x) < 0. Those are x < 0, 0 < x < 2, and 2 < x.
__
4. The difference quotient is defined as ...
dq = (f(x +h) -f(x))/h
The difference quotient is essentially the average slope between (x, f(x)) and (x+h, f(x+h)). That is, it is the slope of the secant line between those two points.
For linear functions, the slope is a constant. The difference quotient is a constant equal to the slope of the line.
Example
f(x) = ax +b . . . . . a linear function with a slope of 'a'
The difference quotient is ...
(f(x+h) -f(x))/h = ((a(x+h)+b) -(ax+b))/h = (ax+ah+b -ax -b)/h = ah/h = a
The difference quotient is the slope of the line.
which polygon will NOT tessellate a plane?
Answer:
pentagons
Step-by-step explanation:
In fact, there are pentagons which do not tessellate the plane. The house pentagon has two right angles. Because those two angles sum to 180° they can fit along a line, and the other three angles sum to 360° (= 540° - 180°) and fit around a vertex.
Answer:
The Regular Pentagon.
Explanation
I got a 100 % on the quiz
Erica’s family is moving away from California. They decided to have a moving sale and sell each item for 70% off the price they originally paid for it. The sofa had an original price of $799, and the love seat had an original price of $549. What is the total cost of both items after the discount?
Find the sale price by multiplying the original price by 70% then add the two prices together to get the total.
799 x 0.70 = 559.30
549 x 0.70 = 384.30
Total: 559.30 + 384.30 = $943.60
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after each of the following.
after 2s : cm2/s
after 5s : cm2/s
after 6s : cm2/s
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Answer:
2s: 45,239 cm²/s5s: 113,097 cm²/s6s: 135,717 cm²/sStep-by-step explanation:
The radius is a function of time:
r(t) = 60t . . . . . radius in cm; time in s
Then the area of the circle is ...
A = πr² = π(60t)² = 3600πt²
The rate of change of area is the derivative of this:
A' = 2·3600πt = 7200πt
The rates of change of interest are ...
after 2s: 45,239 cm²/s
after 5s: 113,097 cm²/s
after 6s: 135,717 cm²/s
A box plot is shown
O
2
4
6
8
10
12
Determine the five-statistical summary of the data. Drag the correct number to each variable in the summary.
14
16
18
20
22 24 26
28
30
Minimum:
Maximum:
Median:
First Quartile:
Third Quartile:
1
2
3
4
11
5
12
6
ما تا ته
13
14
8
21
15
22
16
10
23
17
24
18
25
19
26
20
27
28
29
30
Please answer fast
Answer:
Minimum = 8
Maximum = 28
Median = 22
First Quartile = 12
Third Quartile = 26
Step-by-step explanation:
✔️Minimum value = the value at the beginning of the whisker from your left = 8
✔️Maximum value = the value at the end of the whisker to your right = 28
✔️Median = the value at the vertical line that divides the box into two = 22
✔️First Quartile = the value at the beginning of the edge of the box = 12
✔️Third Quartile = the value at end of the edge of the box = 26
ifteen accounting majors had an average grade of 90 on a finance exam. Seven marketing majors averaged 85, while ten finance majors averaged 93 on the same exam. What is the weighted mean for the 32 students taking the exam? A. 89.84 B. 89.33 C. 89.48 D. Impossible to determine without more information
Answer:
A. 89.84
Step-by-step explanation:
Weighed mean:
Sum of the multiplications of each value by its weight, divided by the sum of the weights.
Weights:
15 had an average of 90.
7 averaged 85.
10 averaged 93.
What is the weighted mean for the 32 students taking the exam?
[tex]M = \frac{15*90 + 7*85 + 10*93}{15 + 7 + 10} = 89.84[/tex]
Thus the correct answer is given by option A.
Find the missing length in the image below
Let it be x
Using basic proportionality theorem
[tex]\\ \sf\longmapsto \dfrac{x}{10}=\dfrac{14}{7}[/tex]
[tex]\\ \sf\longmapsto \dfrac{x}{10}=2[/tex]
[tex]\\ \sf\longmapsto x=10(2)[/tex]
[tex]\\ \sf\longmapsto x=20[/tex]
How to find the inverse of this matrix
[tex]\left[\begin{array}{ccc}1&0\\0&3\\\end{array}\right][/tex]
Answer:
Here we have the matrix:
[tex]M = \left[\begin{array}{ccc}1&0\\0&3\end{array}\right][/tex]
And we want to find its inverse.
The inverse of a 2x2 matrix A is:
(1/det(A))*adj(A)
where det(A) is the determinant of the matrix.
Such that for a matrix:
[tex]A = \left[\begin{array}{ccc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right][/tex]
The determinant is:
det(A) = a₁₁*a₂₂ - a₁₂*a₂₁
in the case of our matrix M, the determinant is:
det(M) = 1*3 - 0*0 = 3
and adj(A) is a transposition along the diagonal, and for the other elements, we just change its sign.
Then for our matrix A we would have:
[tex]adj(A) = \left[\begin{array}{ccc}a_{22}&-a_{12}\\-a_{21}&a_{11}\end{array}\right][/tex]
Then for our matrix M, we have:
[tex]adj(M) = \left[\begin{array}{ccc}3&-0\\-0&1\end{array}\right][/tex]
Then the inverse of the matrix M is:
[tex]M^{-1} = \frac{1}{det(M)} *adj(M) = \frac{1}{3}\left[\begin{array}{ccc}3&0\\0&1\end{array}\right] = \left[\begin{array}{ccc}1&0\\0&1/3\end{array}\right][/tex]
Please i need to find the era bounded by the following curves
Answer:
10 2/3 or 32/ 3
Step-by-step explanation:
5 - x^2 - (2 - 2x) =
= -x^2 + 2x + 3
Integral of (-x^2 + 2x + 3)dx from -1 to 3 =
= -x^3/3 + 2x^2/2 + 3x from -1 to 3
= -x^3/3 + x^2 + 3x from -1 to 3
= -27/3 + 9 + 9 - (1/3 + 1 - 3)
= -9 + 9 + 9 - 1/3 - 1 + 3
= 11 - 1/3
= 10 2/3 = 32/3
Answer:
32/3
Step-by-step explanation:
Check the pdf :)
The scatterplot shows the selling prices of homes and the square feet of living space.
A graph titled home value has square feet (thousands) on the x-axis, and price (hundred thousand dollars) on the y-axis. Points are at (1.2, 1), (1.5, 1.1), (2, 1.5), (2.5, 2). An orange point is at (3.8, 3.9).
Complete the statements based on the information provided.
The scatterplot including only the blue data points shows
✔ a strong positive
correlation. Including the orange data point at (3.8, 3.9) in the correlation would
✔ strengthen
and
✔ increase
the value of
Answer:
✔ a strong positive
✔ strengthen
✔ increase
ED2021
Answer:
- a strong positive
- strengthen
- increase
Which statements describe the data in the bar graph? Check all that apply.
People prefer rock music to any other type of music.
People prefer pop music to any other type of music.
The least favorite genre of music is blues.
The least frequent favorite genre is country.
Four times as many people prefer pop music to blues.
Answer:
People prefer pop music to any other type of music.
The least favorite genre of music is blues.
Four times as many people prefer pop music to blues.
Answer:
People prefer pop music to any other type of music.
The least favorite genre of music is blues.
Four times as many people prefer pop music to blues.
Answer:
B) People prefer pop music to any other type of music.
C) The least favorite genre of music is blues.
E) Four times as many people prefer pop music to blues.
Step-by-step explanation:
edge 2023
Arrange 3/5,5/8,5/6 and 7/4 in ascending order
It’s already in ascending order.
3/5= .6
5/8= .625
5/6= .833
7/4= 1.75
Reggie and Jay go for a walk every morning. Reggie walks 2 14 miles. Jay walks 138 miles less than Reggie. What is the total distance they walk every morning?
Answer:
They walked a total distance of 290 miles every morning.
Step-by-step explanation:
First, we have to subtract 214 and 138.
= Jay walks 76 miles.
Next, we have add 214 and 76.
= 290 mi.
The ratio of total interior angles to total exterior angles of a quadrilateral is
Select one:
a. 3:1
b. 1:2
c. 1:1
d. 2:1
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Answer:
c. 1 : 1
Step-by-step explanation:
The total of exterior angles of any convex polygon is 360°. The total of interior angles of a quadrilateral is 360°. So, the ratio of interest is ...
interior : exterior = 360° : 360° = 1 : 1
the question is in the photo. it is asking for 2 answers
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Answer:
2nd force: 99.91 lbresultant: 213.97 lbStep-by-step explanation:
In the parallelogram shown, angle B is the supplement of angle DAB:
∠B = 180° -77°37' = 102°23'
Angle ACB is the difference of angles 77°37' and 27°8', so is 50°29'.
Now, we know the angles and one side of triangle ABC. We can use the law of sines to solve for the other two sides.
BC/sin(A) = AB/sin(C)
AD = BC = AB·sin(A)/sin(C) = (169 lb)sin(27°8')/sin(50°29') ≈ 99.91 lb
AC = AB·sin(B)/sin(C) = (169 lb)sin(102°23')/sin(50°29') ≈ 213.97 lb
The price of an item has been reduced by 70%. The original price was $30. What is the price of the item now?
Answer:
$9
Step-by-step explanation:
30*(100%-70%)=9
Answer:
9
Step-by-step explanation:
Take the original price
Multiply by the discount percent
30 *70%
30 *.70
21
The discount is 21 percent
Subtract this from the original amount
30-21
9
A cylinder with a base diameter of x units has a volume of excubic units.
Which statements about the cylinder are true,Select
two options.
1)The radius of the cylinder is 2x units.
2)The area of the cylinder's base is 1/4 piex^2square units.
3)The area of the cylinder's base is 1/2 piex^2 square units.
4)The height of the cylinder is 2x units.
5)The height of the cylinder is 4x units.
Answer:3 and 4
Step-by-step explanation:
PLS HELP
If f(x) = x2 -1, what is the equation for f–1(x)?
We have a function,
[tex]f(x)=x^2-1[/tex]
and we are asked to find its inverse function.
An inverse function essentially gets you the original value that was dropped into a function.
For example,
If I put 5 into [tex]f(x)[/tex] I will get 24. Now If I take 24 and put it into the inverse function I have to get number 5 back.
The way to determine the inverse function swap the x and the [tex]f(x)[/tex], then solve for [tex]f(x)[/tex],
[tex]x=f(x)^2-1[/tex]
[tex]f(x)^2=x+1[/tex]
[tex]f(x)=\pm\sqrt{x+1}[/tex]
Of course the notation demands that the obtained function be called,
[tex]f^{-1}(x)=\pm\sqrt{x+1}[/tex]
Hope this helps :)
