Matrix

[tina07]

Hello!
I have 2 questions about matrix:

1. If I need to find a population growth after 25 year. How can I calculate it without a calulator? Every year individually
25 times or there is some other way I should do that?

My matrix is

0 2 3 240
.2 0 0 400
0 .5 0 320


2. For what value of b does (A − I3)x = 0 have a nonzero solution?

0 2 b
.2 0 0
0 .5 0

The book has answer 6, not sure how to get there.

I understand what A- I3 (below), but how to find b value without a determinant, because I did not study determinant yet?

-1 2 b
.2 -1 0
0 .5 -1

Thank you!

 

[HallsofIvy]

Hello!
I have 2 questions about matrix:

1. If I need to find a population growth after 25 year. How can I calculate it without a calulator? Every year individually
25 times or there is some other way I should do that?

My matrix is

0 2 3 240
.2 0 0 400
0 .5 0 320

I have no clue what you are asking here! You say "If I need to find a population growth after 25 year" and give a matrix. What does the one have to do with the other?? Does the matrix show some kind of population growth per year? Are you saying there are four different populations?

2. For what value of b does (A − I3)x = 0 have a nonzero solution?

0 2 b
.2 0 0
0 .5 0

The book has answer 6, not sure how to get there.

I understand what A- I3 (below), but how to find b value without a determinant, because I did not study determinant yet?

-1 2 b
.2 -1 0
0 .5 -1

Thank you!

Then do it directly. If you let \(\displaystyle x= \begin{bmatrix}x \\ y \\ z\end{bmatrix}\), then the equation becomes
\(\displaystyle \begin{bmatrix} -1 & 2 & b \\ .2 & -1 & 0 \\ 0 & .5 & -1 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}\).

Multiply out the left side and set it equal to the right side and you will have three equations in x, y, and z. If you solve that in the usual way, for example, by eliminating until you get down to a single equation in x, something like Px= 0 where P is a formula involving b. The "obvious" solution would be to divide both sides by p getting x= 0 as the only solution. Since you want a non-zero solution, you must not be able to divide by P. And that means P must be equal to 0.

 

[tina07]

Question for #1
A certain colony of squirrels has a life span of less than 3 years. Suppose that the females are divided into three
age groups: those under age 1, those of age 1, and those of age 2. Suppose further that the proportion of newborn
females that survives until age 1 is .2 and that the proportion of one-year-old females that survives until age 2
is .5. Assume also that females under age 1 do not give birth, those of age 1 have, on average, 2 female offspring,
and those of age 2 have, on average, b female offspring.

Suppose there are initially 240 females of age less than
one, 400 of age one, and 320 of age two.

(a) Write a Leslie matrix A for this colony of squirrels.

(b) If b = 3, what will happen to the population in 25
years?

 

[tina07]

2. For what value of b does (A − I3)x = 0 have a nonzero solution?

0 2 b
.2 0 0
0 .5 0

The book has answer 6, not sure how to get there.

I understand what A- I3 (below), but how to find b value without a determinant, because I did not study determinant yet?

-1 2 b
.2 -1 0
0 .5 -1

Am I on the right way?

-x_1+2x_2 +b = 0

0.2x_1=x_2

0.5x_2= x_3

and then
-x_1 + 0.2x_1 +b = 0
b =x_1 - 0.2x_1