Question

in the matrix equation
[a b][a]=[20]
[c d][e] [43]
each variable is positive, an integer, and less than 10. what is a+b+c+d+e?

Answer 1

Where did this question come from?

Answer 2

Are you here cathy?

Answer 3

I think there is no integer solution. Using mathematica, I've

a=2*sqrt(5), b=0, c=-43/(2 sqrt(5)) and d=0. I didn't enter as a matrix...

Answer 4

There is a solution.

Answer 5

You have two equations
a^2 + be = 20
ca + de = 43

Now it's a process of trial and error, and logic, to find values of a,b,c,d,e that satisfy these equations.

I began with different values of a and tried to find matching b and e. Then given those options, you'll quickly realize that those selections are constrained in order that the second equation is true.

It's an unusual question.

Answer 6

also you can use the command solve{{{a,b},{c,d}}.{{a},{e}}={{20},{43}},a,b,c,d,e} in wolframalpha.com ...

Answer 7

Cheating! Much more satisfying to figure it out oneself. ;-)

Also Wolfram has missed at least one answer, the integer valued answer we're looking for.

Answer 8

wolfram gives all 7 possible values of the solution? No integer one included...

Answer 9

anyway, I've to go... Good night :)

Answer 10

sorry...my internet funked.

Answer 11

@asnaseer: don't give the whole answer! This problem is too fun to give all away in one go.

Answer 12

if you look at the 1st equation, we have:\[a^2+be=20\]which leads to:\[
\text{if }a=1\implies a^2=1\implies be=19\text{ reject as 19 is prime and values must be less than 10}\]\[
\text{if }a=2\implies a^2=4\implies be=16\]this leads to "be" being composed of:
16 * 1 (reject as values must be less than 10)
8 * 2 (valid)
4 * 4 (valid)
\[\text{if }a=3\implies a^2=9\implies be=11\text{ reject}\]
\[\text{if }a=4\implies a^2=16\implies be=4\]this leads to "be" being composed of:
4 * 1 (valid)
2 * 2 (valid)

and "a" beyond 4 is not valid as that would lead to a negative "be".

so we get:
a=2 and be=8*2 or 4*4
a=4 and be=4*1 or 2*2

I'm sure we can continue in this vein to solve this problem.

Answer 13

I also really like it that Wolfram doesn't find the answer.

Answer 14

goes to show that we humans still have something to contribute... ;-)

Answer 15

I've been waiting for an example of Wolfram falling down. I didn't expect it here, but there it is.

Answer 16

I've found an even shorter method of arriving at the solution by inferring some more restrictions on the solution set.