Linear Algebra problem. I don\'t know how to approach this kind of problem so pl

Question

Linear Algebra problem. I don't know how to approach this kind of problem so please help.

Sorry that there are many small questions, please answer as compelete as possible. I will give a thumb up, thank you!

(1) Consider a matrix of the form and suppose that 2) The first row sums to 4, (3) The second row sums to 8, 4) The first lmn sums to 2, (5) The second column sums to some unknown value s It turns out that there is only one value of s that allows the other three conditions but for that value of s there are an infinite number of matrices that work. Write the four conditions as an augmented system and use techniques from sections 1.1 and 1.2 to determine s Then, find three such matrices that satisfy all four row- and column-sum conditions to be satisfied,

Explanation / Answer

Since there are 5 variables (a,b,c,d,s), there will be either no solutions or infinite solutions.

write all the equation one by one

(2)

a + b = 4

(3)

c+d = 8

(4)

a + c = 2

(5)

b + d = s

put the value of

a + b = 4...(1)

c+d = 8...(2)

a + c = 2...(3)  

b + d = s...(4)

eq(3) - eq(1)

b - c = 2

b = c+2

eq(2) - eq(3)

d-a = 6

d = a+6 = 4 - b + 6 = 10 - b

put in the eq(4)

10 - b + b = s

s = 10

put both b and d in form of c

1) pick b and d in such a way that b + d = 10, so let's pick b = 5, d = 5

so c = b - 2 =5 - 2 = 3

a = 4 - b = -1

so answer is [(a,b) (c,d)] = [(-1,5) (3,5]

2) now let's pick b = 4, d = 6

so c = b - 2 =4 - 2 = 2

a = 4 - b = 0

so answer is [(a,b) (c,d)] = [(0,4) (2,6]

3) now let's pick b = 3, d = 7

so c = b - 2 =3 - 2 = 1

a = 4 - 3 = 1

so answer is [(a,b) (c,d)] = [(1,3) (1,7]

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