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प्रश्न
The sum of three numbers is 6. If we multiply third number by 3 and add it to the second number we get 11. By adding the first and third number we get a number which is double the second number. Use this information and find a system of linear equations. Find the three numbers using matrices.
उत्तर
Let the three numbers be x, y and z respectively.
According to the first condition,
x + y + z = 6
According to the second condition,
3z + y = 11 i.e., y + 3z = 11
According to the third condition,
x + z = 2y i.e., x – 2y + z = 0
Matrix form of the above system of equations is
`[(1, 1, 1),(0, 1, 3),(1, -2, 1)] [(x),(y),(z)] = [(6),(11),(0)]`
Applying R2 ↔ R3, we get
`[(1, 1, 1),(1, -2, 1),(0, 1, 3)] [(x),(y),(z)] = [(6),(0),(11)]`
Applying R2 → R2 – R1, we get
`[(1, 1, 1),(0, -3, 0),(0, 1, 3)] [(x),(y),(z)] = [(6),(-6),(11)]`
Applying R3 → 3R3 + R2, we get
`[(1, 1, 1),(0, -3, 0),(0, 0, 9)] [(x),(y),(z)] = [(6),(-6),(27)]`
Hence, the original matrix is reduced to an upper triangular matrix.
∴ `[(x + y + z),(0 - 3y + 0),(0 + 0 + 9z)][(6),(-6),(27)]`
∴ By equality of martices, we get
x + y + z = 6 ......(i)
– 3y = – 6 ......(ii)
i.e., y = 2
9z = 27 ......(iii)
i.e., z = 3
Substituting y = 2 and z = 3 in equation (i), we get
x + 2 + 3 = 6
∴ x = 1
∴ 1, 2 and 3 are the required numbers.
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