All categories

3 years ago

Give an example of a 2x2 matrix without any real eigenvalues:___________

Mathematics

1 answer:

9966 [12]3 years ago

3 0

Answer:

Step-by-step explanation:

An eigenvalue of n × n is a function of a scalar $\lambda$ considering that there is a solution (i.e. nontrivial) to an eigenvector x of Ax =

Suppose the matrix $A = \left[\begin{array}{cc}-1&-1\\2&1\\ \end{array}\right]$

Thus, the equation of the determinant (A - $\lambda$ 1) = 0

This implies that:

$\left[\begin{array}{cc}-1-\lambda &-1\\2&1- \lambda\\ \end{array}\right] =0$

$-(1 - \lambda^2 ) + 2 = 0$

$-1 + \lambda ^2 + 2= 0$

$\lambda^2 +1 =0$

Hence, the eigenvalues of the equation are $\mathtt{\lambda = i , -i}$

Also, the eigenvalues can be said to be complex numbers.

You might be interested in

How many decimals can 2/5 represent?

Alexus [3.1K]

2/5 in decimal form is 0.4.

Hope this helps!

5 0

3 years ago

Each candle uses 10 ounces of wax and 1 ounces of scent. Explain in complete sentences how I can determine the number of candles

Sphinxa [80]

You would multiply 50x16 being 16 ounces= 1 pound and than divide that by 10

4 0

3 years ago

Someone help me with this still im stock on it plz

pav-90 [236]

Answer:

a>4 or a < 8

Step-by-step explanation:

We can start with the inequality on the left

-a + 2(4+3a) > 28

expand using the distributive property

-a + 8 + 6a > 28

5a + 8 > 28

subtract 8 from both sides to isolate the variable and its coefficient

5a > 20

divide both sides by 5 to isolate a

a > 4

moving on to the inequality on the right

2a - (3a+4) > -12

treat the minus sign as a -1

2a + (-1)(3a+4) > - 12

expand

2a - 3a - 4 > - 12

-a - 4 > -12

add 4 to both sides to isolate a and its coefficient

-a > - 8

multiply both sides by -1 but change the sign because we are multiplying by a negative number

a < 8

5 0

2 years ago

A photoconductor film is manufactured at a nominal thickness of 25 mils. The product engineer wishes to increase the mean speed

AURORKA [14]

Answer:

A 98% confidence interval estimate for the difference in mean speed of the films is [-0.042, 0.222].

Step-by-step explanation:

We are given that Eight samples of each film thickness are manufactured in a pilot production process, and the film speed (in microjoules per square inch) is measured.

For the 25-mil film, the sample data result is: Mean Standard deviation 1.15 0.11 and For the 20-mil film the data yield: Mean Standard deviation 1.06 0.09.

Firstly, the pivotal quantity for finding the confidence interval for the difference in population mean is given by;

P.Q. = $\frac{(\bar X_1 -\bar X_2)-(\mu_1- \mu_2)}{s_p \times \sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }$ ~ $t__n_1_+_n_2_-_2$

where, $\bar X_1$ = sample mean speed for the 25-mil film = 1.15

$\bar X_1$ = sample mean speed for the 20-mil film = 1.06

$s_1$ = sample standard deviation for the 25-mil film = 0.11

$s_2$ = sample standard deviation for the 20-mil film = 0.09

$n_1$ = sample of 25-mil film = 8

$n_2$ = sample of 20-mil film = 8

$\mu_1$ = population mean speed for the 25-mil film

$\mu_2$ = population mean speed for the 20-mil film

Also, $s_p =\sqrt{\frac{(n_1-1)s_1^{2}+ (n_2-1)s_2^{2}}{n_1+n_2-2} }$ = $\sqrt{\frac{(8-1)\times 0.11^{2}+ (8-1)\times 0.09^{2}}{8+8-2} }$ = 0.1005

Here for constructing a 98% confidence interval we have used a Two-sample t-test statistics because we don't know about population standard deviations.

So, 98% confidence interval for the difference in population means, ( $\mu_1-\mu_2$ ) is;

P(-2.624 < $t_1_4$ < 2.624) = 0.98 {As the critical value of t at 14 degrees of

freedom are -2.624 & 2.624 with P = 1%}

P(-2.624 < $\frac{(\bar X_1 -\bar X_2)-(\mu_1- \mu_2)}{s_p \times \sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }$ < 2.624) = 0.98

P( $-2.624 \times {s_p \times \sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }$ < $2.624 \times {s_p \times \sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }$ < ) = 0.98

P( $(\bar X_1-\bar X_2)-2.624 \times {s_p \times \sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }$ < ( $\mu_1-\mu_2$ ) < $(\bar X_1-\bar X_2)+2.624 \times {s_p \times \sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }$ ) = 0.98

98% confidence interval for ( $\mu_1-\mu_2$ ) = [ $(\bar X_1-\bar X_2)-2.624 \times {s_p \times \sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }$ , $(\bar X_1-\bar X_2)+2.624 \times {s_p \times \sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }$ ]

= [ $(1.15-1.06)-2.624 \times {0.1005 \times \sqrt{\frac{1}{8}+\frac{1}{8} } }$ , $(1.15-1.06)+2.624 \times {0.1005 \times \sqrt{\frac{1}{8}+\frac{1}{8} } }$ ]

= [-0.042, 0.222]

Therefore, a 98% confidence interval estimate for the difference in mean speed of the films is [-0.042, 0.222].

Since the above interval contains 0; this means that decreasing the thickness of the film doesn't increase the speed of the film.

7 0

2 years ago

Would the scale factor 4.2 enlarge,reduce,or preserve

Juli2301 [7.4K]

Hey there! I'm happy to help!

Let's look at what scale factors would be able to enlarge, reduce, and preserve first. We will use the point (1,1) as an example.

ENLARGE

When we talk about scale factors, both of our numbers of the ordered pair will be multiplied by that number. To make it bigger, we will multiply it by anything larger than a one, we could have a scale factor such as 2, 100, 579395. etc.!

Dilation of 2

(1,1)⇒(2,2)

REDUCE

To reduce our ordered pair, our scale factor must be a number greater than zero but less than one because it cannot switch quadrants, which is what would happen if we used negative numbers. Our reduced number must be greater than zero. We can use 1/2, 1/4, 1/100, 0.67 etc.

Dilation of 1/2

(1,1)⇒(0.5,0.5)

PRESERVE

What would we multiply an ordered pair by to make it stay the same? One of course! This is the only number that will keep it the same!

Dilation of 1

(1,1)⇒(1,1)

Now, let's think of any scale factor as X and see what rules it has to follow to be enlarging, reducing, or preserving.

ENLARGING SCALE FACTOR: X>1

REDUCING SCALE FACTOR: 0<X<1

PRESERVING SCALE FACTOR: X=1

So, where does 4.2 fall? Well, as you can see, 4.2 is greater than one, so this scale factor would make our point larger.

Therefore, 4.2 would enlarge.

I hope that this helps! Have a wonderful day!

8 0

3 years ago