NO LINKS OR ANSWERING QUESTIONS YOU DON'T KNOW!! Part 2.

2. What is a determinant and what role does it play with matrices (Hint: What does a determinant of 0 mean)? How can this be used when solving systems of equations?

9514 1404 393

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:

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.

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

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

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

Please i need to find the era bounded by the following curves

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?

If f:X is 3x + b and ff(2) = 12, find the value of b​

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?

which polygon will NOT tessellate a plane?

how to graph quadratic relationship for h(x)=(x-1)^2-9

Find the length of AC on this triangle

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

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.

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

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

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

I need help guys thanks so much

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.

PLS HELP
If f(x) = x2 -1, what is the equation for f–1(x)?

Identify the decimals labeled with letters A B and C on the scale

How to find the inverse of this matrix
[tex]\left[\begin{array}{ccc}1&0\\0&3\\\end{array}\right][/tex]

The price of an item has been reduced by 70%. The original price was $30. What is the price of the item now?​

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?

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

Does the point (0, 0) satisfy the equation y = 9x?

Arrange 3/5,5/8,5/6 and 7/4 in ascending order

the question is in the photo. it is asking for 2 answers

(a) 4x + 3y + 2x + 7y

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.

What is the vertex of the graph of this function
y= -(x+2) (x+4)

Find the missing length in the image below