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Expansion Numerically Impractical. Show that the computation of an nth-order determinant by expansion involves multiplications,

posledela

Answer:

number of multiplies is n!
n=10, 3.6 ms
n=15, 21.8 min
n=20, 77.09 yr
n=25, 4.9×10^8 yr

Step-by-step explanation:

Expansion of a 2×2 determinant requires 2 multiplications. Expansion of an n×n determinant multiplies each of the n elements of a row or column by its (n-1)×(n-1) cofactor determinant. Then the number of multiplies is ...

mpy[n] = n·mp[n-1]

mpy[2] = 2

So, ...

mpy[n] = n! . . . n ≥ 2

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If each multiplication takes 1 nanosecond, then a 10×10 matrix requires ...

10! × 10^-9 s ≈ 0.0036288 s ≈ 0.004 s . . . for 10×10

Then the larger matrices take ...

n=15, 15! × 10^-9 ≈ 1307.67 s ≈ 21.8 min

n=20, 20! × 10^-9 ≈ 2.4329×10^9 s ≈ 77.09 years

n=25, 25! × 10^-9 ≈ 1.55112×10^16 s ≈ 4.915×10^8 years

_____

For the shorter time periods (less than 100 years), we use 365.25 days per year.

For the longer time periods (more than 400 years), we use 365.2425 days per year.

8 0

4 years ago

Given the system of linear equations.

Karo-lina-s [1.5K]

Part A:

x + y = 9

4x + y = 6

To solve this system using substitution, begin by isolating either x or y in the first equation. I'll isolate x.

x + y = 9

x = 9 - y

Substitute this expression (9 - y) into the second equation for x and solve.

4x + y = 6

4(9 - y) + y = 6

36 - 4y + y = 6

36 - 3y = 6

-3y = -30

y = 10

To find x, substitute 10 for y into either of the original equations.

x + y = 9

x + 10 = 9

x = -1

Finally, check all work by substituting the x- and y-values into each original equation.

x + y = 9 --> -1 + 10 = 9 --> 9 = 9 --> True

4x + y = 6 --> 4(-1) + 10 = 6 --> -4 + 10 = 6 --> 6 = 6 --> True

The answer for Part A is x = -1 and y = 10; (-1, 10).

Part B:

For graphing, it's easier to get the equations into slope-intercept form. Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. To get our equations into slope-intercept form, we must simply isolate y.

x + y = 9

y = 9 - x

y = -x + 9

4x + y = 6

y = 6 - 4x

y = -4x + 6

Now that we have our slope-intercept equations, we can easily graph them, since their slopes and y-intercepts are readily visible.

Let's start with the y-intercepts. They are (0, 9) and (0, 6). You can plot those points on the graph.

Now, the slopes. The slope of the first line is -1, this means it declines. To plot this, start where you plotted the y-intercept, count one unit down, and then one unit to the right, and plot that point. Continue doing that and connect the dots, and you will have plotted the first line. The slope of the second line is -4, so it also declines. For this line, count four units down, and then one to the right and plot that point. Likewise, continue this and connect the coordinates, and you will have your line. (See attachment for graph.)

The lines do indeed intersect at (-1, 10); our answer is verified by graphing.