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# If Three Numbers Are Added, Their Sum is 2. If Two Times the Second Number is Subtracted from the Sum of - Mathematics and Statistics

Sum

If three numbers are added, their sum is 2. If two times the second number is subtracted from the sum of first and third numbers we get 8 and if three times the first number is added to the sum of second and third numbers we get 4. Find the numbers using matrices.

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#### Solution

Let the three numbers x , y , z.

From given condition, we have

x + y + z = 2  .......(1)

x + z - 2y = 8

x - 2y + z = 8  ......(2)

And

3x + y + z = 4   .....(3)

Given all equation can be written in matrix form as ,

[(1,1,1),(1,-2,1),(3,1,1)] [(x),(y),(z)] = [(2),(8),(4)]

Consider , AX = B

On multiplying A-1 both sides , we get

A-1 . A.X = A-1.B

X = A-1 . B   ......(4)

Now

A-1 = 1/|"A"|  "Adj (A)"

|A| = |(1,1,1),(1,-2,1),(3,1,1)|

|A| = 6

Now , For Adj (A) , we need minors and co-factors.

M11 = -3 , M12 = -2 , M13 = 7

M21 = 0 , M22 = -2 , M23 = -2

M31 = 3 , M32 = 0 ,M33 = -3

Therefore ,

A-1 = 1/6 [(-3,0,3),(2,-2,0),(7,2,-3)]

From equation (4),

X = 1/6 [(-3,0,3),(2,-2,0),(7,2,-3)] [(2),(8),(4)]

[(x),(y),(z)] = [(1),(-2),(3)]

Hence , x = 1 , y = -2 , z = 3

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