Why cant you take the inverse of a matrix with det = 0?
A matrix with a determinant of zero is called a singular matrix and doesn't have an inverse because it indicates the system of equations it represents doesn't have a unique solution, and the transformation it performs is not reversible.
When a matrix has a determinant of zero (det A = 0), it is classified as a singular matrix and is not invertible. This characteristic implies that the matrix cannot be used to find unique solutions for a set of linear equations because such a matrix corresponds to a system of equations that has either no solution or an infinite number of solutions.
The ability to have an inverse matrix is crucial as it allows us to solve matrix equations and essentially 'undo' the transformations applied by the original matrix.
The reason why a zero determinant indicates the absence of an inverse lies in the mathematics of linear transformations.
A determinant of zero suggests that the transformation associated with the matrix collapses the dimensionality of the space, which means some information about the original vectors is lost, and hence, an inverse operation to recover the original vectors cannot exist.
To further illustrate this, consider the matrix equation AB = I, where A is our original matrix and B is its supposed inverse yielding the identity matrix I. It follows from the properties of determinants that det(AB) = det(A) x det(B).
If det(A) is zero, then the product det(A) x det(B) will also be zero, not equal to 1, which is the determinant of the identity matrix. Therefore, B cannot serve as the inverse of A.
In other words, having a non-zero determinant is a prerequisite for a matrix to have an inverse, as it ensures that the system of equations it represents is solvable and that the matrix transformation is reversible.
In a triangle, the measure of the first angle is three timesthree times the measure of the second angle. the measure of the third angle is 80 degrees80° more than the measure of the second angle. use the fact that the sum of the measures of the three angles of a triangle is 180degrees° to find the measure of each angle.
The sum of the two numbers is 50 and their difference is 4 what are the two numbers
Two numbers are between 20 and 30. their greatest common factor is 4.Which two numbers could they be?
11/12+(−7/12). Write your answer as a fraction in simplest form.
The result of the operation 11/12 + (-7/12) simplifies to 1/3, because the sum of the numerators is 4, and we keep the same denominator of 12, giving us 4/12. In simplest form, this fraction is 1/3.
Explanation:The question is asking us to perform an addition operation on two fractions. Specifically, we are asked to add 11/12 and -7/12. When adding or subtracting fractions, the most important thing is that they have the same denominator, which our fractions do. Here's how you perform the operation:
Add the numerators of the fractions: 11 + (-7) = 4.
You keep the same denominator: 12.
So, 11/12 + (-7/12) = 4/12.
To simplify this fraction, we find the greatest common divisor of 4 and 12, which is 4, and divide both the numerator and denominator by it. So, 4/12 simplifies to 1/3.
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Rectangle EFGH is graphed on a coordinate plane with vertices at E(-3,5), F(6,2), G(4,-4), and H(-5,-1). Find the slopes of each side. What do you notice about the slopes of opposite sides? What do you notice about the slopes of adjacent sides?
Two slices of Dans Famous pizza have 230 calories how many calories would you expect to be in 5 slices of the same pizza
Point G is between points F and H. FH = 102, FG = 5x + 9, and HG = 9x − 5. Show your work.
A.) What is the value of x ?
B.) What is the length of ̅̅̅̅FG?
C.) What is HG?
The steps to do it. It says "solve for x. Each figure is a parallelogram. Show your work.
The value of x is 4.
To solve for x, we'll use the property of parallelograms that opposite angles are supplementary.
Given that one angle is [tex]\(140^\circ\)[/tex], we can set up the equation:
140 + 10x = 180
Now let's solve for x:
10x = 180 - 140
10x = 40
[tex]x = \frac{40}{10}[/tex]
x = 4
Therefore, the value of x is 4. Adjacent angles in a parallelogram are indeed supplementary.
What does it mean if two angles are congruent?
Pqr is a right angle. If pq=8 what is equivalent
What would be an appropriate measure to describe the depth of a lake?
miles
cubic centimeters
milliliters
feet
Answer:
Feet is the appropriate unit.
Step-by-step explanation:
Feet will be an appropriate measure to describe the depth of a lake.
Miles is a very large unit usually used to describe distance between two places.
Cubic centimeters is a unit of volume.
Millimeters is a very small unit used to describe small objects like the diameter of a penny etc.
Therefore, feet is the most appropriate unit to measure the depth of the lake.
Find the least common multiple of 3,4,5,6,10,15
Calculus: Help ASAP
Evaluate exactly the value of the integral from negative 1 to 0 of the product of the cube of the quantity 4 times x to the 6th power plus 2 times x and 12 times x to the 5th power plus 1, dx. Your work must include the use of substitution and the antiderivative.
Answer:
2.264 (3 d.p.)
Step-by-step explanation:
Given integral:
[tex]\displaystyle \int^0_{-1} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x[/tex]
First, evaluate the indefinite integral using the method of substitution.
[tex]\textsf{Let} \;\;u = 4x^6+2x[/tex]
Find du/dx and rewrite it so that dx is on its own:
[tex]\dfrac{\text{d}u}{\text{d}x}=24x^5+2 \implies \text{d}x=\dfrac{1}{24x^5+2}\; \text{d}u[/tex]
Rewrite the original integral in terms of u and du, and evaluate:
[tex]\begin{aligned}\displaystyle \int\left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x&=\int \left(u\right)^3\left(12x^5+1\right)\cdot \dfrac{1}{24x^5+2}\; \text{d}u\\\\&=\int \left(u\right)^3\left(12x^5+1\right)\cdot \dfrac{1}{2(12x^5+1)}\; \text{d}u\\\\&=\int \dfrac{u^3\left(12x^5+1\right)}{2(12x^5+1)}\; \text{d}u\\\\&=\displaystyle \int \dfrac{u^3}{2}\; \text{d}u\\\\&=\dfrac{u^{3+1}}{2(3+1)}+C\\\\&=\dfrac{u^4}{8}+C\end{aligned}[/tex]
Substitute back u = 4x⁶ + 2x:
[tex]=\dfrac{(4x^6+2x)^4}{8}+C[/tex]
Therefore:
[tex]\displaystyle \int \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x=\dfrac{(4x^6+2x)^4}{8}+C[/tex]
To evaluate the definite integral, we must first determine any intervals within the given interval -1 ≤ x ≤ 0 where the curve lies below the x-axis. This is because when we integrate a function that lies below the x-axis, it will give a negative area value.
Find the x-intercepts by setting the function to zero and solving for x.
[tex]\left(4x^6+2x\right)^3\left(12x^5+1\right)=0[/tex]
Therefore:
[tex]\begin{aligned}\left(4x^6+2x\right)^3&=0\\4x^6+2x&=0\\x(4x^5+2)&=0\end{aligned}[/tex]
[tex]x=0[/tex]
[tex]\begin{aligned}4x^5+2&=0\\4x^5&=-2\\x^5&=-\frac{1}{2}\\x&=\sqrt[5]{-\dfrac{1}{2}}\end{aligned}[/tex]
[tex]\begin{aligned}12x^5+1&=0\\12x^5&=-1\\x^5&=-\dfrac{1}{12}\\x&=\sqrt[5]{-\dfrac{1}{12}}\end{aligned}[/tex]
Therefore, the curve of the function is:
Below the x-axis between -1 and ⁵√(-1/2).Above the x-axis between ⁵√(-1/2) and ⁵√(-1/12).Below the x-axis between ⁵√(-1/12) and 0.So to calculate the total area, we need to calculate the positive and negative areas separately and then add them together, remembering that if you integrate a function to find an area that lies below the x-axis, it will give a negative value.
Integrate the function between -1 and ⁵√(-1/2).
As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:
[tex]\begin{aligned}A_1&=-\displaystyle \int_{-1}^{\sqrt[5]{-\frac{1}{2}}} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=-\left[\dfrac{(4x^6+2x)^4}{8}\right]_{-1}^{\sqrt[5]{-\frac{1}{2}}}\\\\&=-\left[\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{2}}\right)^6+2\left(\sqrt[5]{-\frac{1}{2}}\right)\right)^4}{8}\right)-\left(\dfrac{(4(-1)^6+2(-1))^4}{8}\right)\right]\\\\&=-[0-2]\\\\&=2\end{aligned}[/tex]
Integrate the function between ⁵√(-1/2) and ⁵√(-1/12).
[tex]\begin{aligned}A_2&=\displaystyle \int_{\sqrt[5]{-\frac{1}{2}}} ^{\sqrt[5]{-\frac{1}{12}}} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=\left[\dfrac{(4x^6+2x)^4}{8}\right]_{\sqrt[5]{-\frac{1}{2}}}^{\sqrt[5]{-\frac{1}{12}}}\\\\&=\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{12}}\right)^6+2\left(\sqrt[5]{-\frac{1}{12}}\right)\right)^4}{8}\right)-\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{2}}\right)^6+2\left(\sqrt[5]{-\frac{1}{2}}\right)\right)^4}{8}\right)\\\\\end{aligned}[/tex]
[tex]\begin{aligned}&=\dfrac{625}{648\sqrt[5]{12^4}}-0\\\\&=0.132117398...\end{aligned}[/tex]
Integrate the function between ⁵√(-1/12) and 0.
As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:
[tex]\begin{aligned}A_3&=-\displaystyle \int_{\sqrt[5]{-\frac{1}{12}}}^0 \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=-\left[\dfrac{(4x^6+2x)^4}{8}\right]_{\sqrt[5]{-\frac{1}{12}}}^0\\\\&=-\left[\left(\dfrac{(4(0)^6+2(0))^4}{8}\right)-\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{12}}\right)^6+2\left(\sqrt[5]{-\frac{1}{12}}\right)\right)^4}{8}\right)\right]\\\\&=-\left[0-\dfrac{625}{648\sqrt[5]{12^4}}\right]\\\\&=\dfrac{625}{648\sqrt[5]{12^4}}\\\\&=0.132117398...\\\\\end{aligned}[/tex]
To evaluate the definite integral, sum A₁, A₂ and A₃:
[tex]\begin{aligned}\displaystyle \int^0_{-1} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x&=2+2\left( \dfrac{625}{648\sqrt[5]{12^4}}\right)\\\\&=2+ \dfrac{625}{324\sqrt[5]{12^4}}\right}\\\\&=2.264\; \sf (3\;d.p.)\end{aligned}[/tex]
The rate of change is constant in each table. Find the rate of change. Explain what the rate of change means for the situation. time (hours) 4, 6, 8, 10 distance (miles) 212, 318, 424, 530
Answer:
The car travels 53 miles per hour.
Step-by-step explanation:
time (hours) 4 6 8 10
distance (miles) 212 318 424 530
The rate of change can be given as:
[tex]\frac{318-212}{6-4}[/tex] = [tex]\frac{106}{2}[/tex] = 53 mph
[tex]\frac{424-318}{8-6}[/tex] = [tex]\frac{106}{2}[/tex] = 53 mph
Hence, the rate of change is 53 mph or we can say the car travels 53 miles per hour.
The sum of 4 consecutive even integers is 36, what is the 3rd largest integer in the set?
The sum of four consecutive even integers that equal 36 is broken down to find the smallest integer. Once the smallest integer is found (6), we can determine that the third largest integer in the sequence is 8.
Explanation:To solve for the third largest integer in a set of four consecutive even integers that sum to 36, we can define the smallest integer as x. The next integers would be x + 2, x + 4, and x + 6. Setting up the equation:
x + (x + 2) + (x + 4) + (x + 6) = 36
Combining like terms, we get:
4x + 12 = 36
Subtracting 12 from both sides:
4x = 24
Dividing by 4:
x = 6
Now we have our integers: 6, 8, 10, 12. The third largest integer, which is the second smallest, is 8.
what is the measure on angle A?
- 110
-70
250
55
What is the sum of the first five terms of the geometric series 3 − 6 + 12 − . . . ?
3,−6,12,−24,...
You can rewrite this as:
(−1)n3⋅(1,2,4,8,...)
If we focus on 1,2,4,8...
an+1=2an
with a0=1.
This can be written out as a nice sum:
N∑n=02n=20+21+22+23+...
=1+2+4+8+...
Thus, now we can recombine everything to get:
3N∑n=0(−1)n2n
The parent function of a graph is f(x) = x2. The graph shifts a units to the left and down b units. Which function models the transformed function?
A) y = x2 - a + b
B) y = (x - a)2 - b
C) y = (x + a)2 - b
D) y = (x - b)2 - a
Answer:
Option c
Step-by-step explanation:
A parent function is given by f(x) = x²
If the graph of the parent quadratic function is shifted a units to the left then new function will become
f'(x) = (x+a)²
Now the new function f'(n) is shifted b units down then the shifted function will be
f" (x) = (x + a)² - b
Therefore, Option c. is the answer.
In sakura's garden for every 5 red flowers, there are 10 yellow flowers. There are a total of 75 yellow and red flowers in her garden. How maNY red flowers are in sakura'sgarden?
Answer:
25 red flowers
Step-by-step explanation:
yeah
The length of a rectangle is four more than five times its width. its perimeter is 4444 inches. find its dimensions (length and width).
The length of a rectangular garden is 3yd more than twice it’s width. The perimeter of the garden is 36yd. What are the width and length of the garden?
Why do two negative numbers multiplied equal a positive?
which of the following equations represents a proportional relationship? Choose all that apply.
A. x=2y
B.a = 1/3b
C.4x=y-2
D.7/3x=n
E.1/2=1/4
The options A, B, and D have a proportional relationship.
What is a proportional relationship?It is the relationship between two variables where their ratios are equivalent.
Consider the first equation, x = 2y.
It can be written like the ratio of x and y.
i.e [tex]\frac{x}{y} -\frac{2}{1}[/tex]
So, option A is proportional at x = 2 and y = 1.
For option B, It can rewrite as
[tex]\frac{a}{b} =\frac{1}{3}[/tex]
So, option B is proportional when a = 1 and b = 3.
Option C is not proportional since it can't be written as a ratio.
Option D can be written as
[tex]\frac{7}{3} =\frac{n}{x}[/tex]
It is proportional when n = 7 and x = 3.
Option E is not proportional since it is not equal.
Therefore the proportional options are A,B, and D.
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Combine like terms.
9 + 3x – 9x + 16
A.19x
B.3x + 16
C.25 – 6x
D.18x + 3x + 16
Can someone help me with this question?
14. For the equation 5x + 36 = x, which value could be a solution? A)–9 B) 5 C)9 D)–5
If you horizontally stretch the quadratic parent function, f(x) = x2, by a factor of 4, what is the equation of the new function?
A. g(x) =1/4 x2
B. g(x) = (1/4x)2
C. g(x) = 4x2
D. g(x) = (4x)2
Answer:
B is your answer
Step-by-step explanation:
g(x)=(1/4x)^2
Find the coordinates of the midpoint of the segment whose endpoints are given . e(4,-4) , f (1,7)
does the set of numbers form a pythagorean triple explain 24 , 10 ,26