MATHEMATICS COLLEGE

consider the quadratic form q(x,y,z)=11x^2-16xy-y^2+8xz-4yz-4z^2. Find an orthogonal change of variable that eliminates the cross product in q(x,y,z) and express q in the new variables.

Answers

Answer 1

Answer:

Answer:

$q(x,y,z)=16x^(2)-5y^(2)-5z^(2)$

Step-by-step explanation:

The given quadratic form is of the form

$q(x,y,z)=ax^2+by^2+dxy+exz+fyz$ .

Where $a=11,b=-1,c=-4,d=-16,e=8,f=-4$ .Every quadratic form of this kind can be written as

$q(x,y,z)={\bf x}^(T)A{\bf x}=ax^2+by^2+cz^2+dxy+exz+fyz=\left(\begin{array}{ccc}x&y&z\end{array}\right) \left(\begin{array}{ccc}a&(1)/(2) d&(1)/(2) e\n(1)/(2) d&b&(1)/(2) f\n(1)/(2) e&(1)/(2) f&c\end{array}\right) \left(\begin{array}{c}x&y&z\end{array}\right)$

Observe that $A$ is a symmetric matrix. So $A$ is orthogonally diagonalizable, that is to say, $D=Q^(T)AQ$ where $Q$ is an orthogonal matrix and $D$ is a diagonal matrix.

In our case we have:

$A=\left(\begin{array}{ccc}11&((1)/(2))(-16) &((1)/(2)) (8)\n((1)/(2)) (-16)&(-1)&((1)/(2)) (-4)\n((1)/(2)) (8)&((1)/(2)) (-4)&(-4)\end{array}\right)=\left(\begin{array}{ccc}11&-8 &4\n-8&-1&-2\n4&-2&-4\end{array}\right)$

The eigenvalues of $A$ are $\lambda_(1)=16,\lambda_(2)=-5,\lambda_(3)=-5$ .

Every symmetric matriz is orthogonally diagonalizable. Applying the process of diagonalization by an orthogonal matrix we have that:

$Q=\left(\begin{array}{ccc}(4)/(√(21))&-(1)/(√(17))&(8)/(√(357))\n(-2)/(√(21))&0&\sqrt{(17)/(21)}\n(1)/(√(21))&(4)/(√(17))&(2)/(√(357))\end{array}\right)$

$D=\left(\begin{array}{ccc}16&0&0\n0&-5&0\n0&0&-5\end{array}\right)$

Now, we have to do the change of variables ${\bf x}=Q{\bf y}$ to obtain

$q({\bf x})={\bf x}^(T)A{\bf x}=(Q{\bf y})^(T)AQ{\bf y}={\bf y}^(T)Q^(T)AQ{\bf y}={\bf y}^(T)D{\bf y}=\lambda_(1)y_(1)^(2)+\lambda_(2)y_(2)^(2)+\lambda_(3)y_(3)^(2)=16y_(1)^(2)-5y_(2)^(2)-5y_(3)^2$

Which can be written as:

$q(x,y,z)=16x^(2)-5y^(2)-5z^(2)$

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How to work 48÷(4+2×10)-1

Answers

Answer:

The answer is 1

Step-by-step explanation:

In order to do this equation you use PEMDAS Parentheses, Exponents, Multiplication , Division, Addition and Subtraction

So,

$48 / (4+2X10)-1\n 48 /(4+20)-1\n 48/(24)-1\n48/24-1\n2-1\n1$

Hoped this helped :D

Solve for the portion. Round to hundredths when necessary.What is the portion if the base is 1,850 and the rate is 150%?

Answers

The portion is $2,475 if the base is 1,850 and the rate is 150%.

What are Arithmetic operations?

Arithmetic operations can also be specified by subtracting, dividing, and multiplying built-in functions. The operator that performs the arithmetic operation is called the arithmetic operator.

* Multiplication operation: Multiplies values on either side of the operator

For example 12*2 = 24

We have to determine the portion if the base is 1,850.

Here rate is 150% or 150/100 = 1.50.

The base = 1,650.

Since the portion is the multiplication of base and rate.

The portion = base × rate.

The portion = 1650 × 1.50.

The portion = $2,475

Therefore, the portion is $2,475 if the base is 1,850 and the rate is 150%.

Answer:

portion = base * rate. rate is 150% or 150/100 = 1.50. base = 1,650. portion = 1650*1.50. portion = $2,475

Step-by-step explanation:

plz mark brainliest!!!!!!!!!!!!!!!!!!!!!!!!

Which expression is equivalent to (m2p–2r7)(m–5p4r2)?

Answers

Answer:

=(m2p+−2r7)(m+−5p4r2)

=(m2p)(m)+(m2p)(−5p4r2)+(−2r7)(m)+(−2r7)(−5p4r2)

=m3p−5m2p5r2−2mr7+10p4r9

=10p4r9−5m2p5r2−2mr7+m3p

Step-by-step explanation:

I NEED YOUR HELP 22 POINTSA cone has a base of 30 in^2 and a height of 8.8 what is the volume of the cone?
Plz show how to do it I'm stuck

Answers

Hello from MrBillDoesMath!

Answer:

88 in^3

Discussion:

The volume of a cone of height h and base radius r is

Pi * R^2 * (h/3)

But Pi * r^2 ( = 30) is the area of the base so the volume is:

(30) * (8.8)/3 =

(30/3) ( 8.8) =

10 * 8.8 =

88 in^3

Thank you,

MrB

Answer:

8293.8in

Step-by-step explanation:

Please help!! Thank youu

Answers

Answer:

Your answer is C

Step-by-step explanation:

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Answers

Answer:

This means that there is a 95% probability that the confidence interval will contain the true population mean. Thus, P( [sample mean] - margin of error < μ < [sample mean] + margin of error) = 0.95.

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