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Question
Express the following equations in matrix form and solve them by method of reduction.
x + y + z = 1, 2x + 3y + 2z = 2 and x + y + 2z = 4
Solution
Matrix form of the given system of equations is
`[(1, 1, 1),(2, 3, 2),(1, 1, 2)][(x), (y), (z)] = [(1), (2), (4)]`
This is of the for AX = B,
where A = `[(1, 1, 1),(2, 3, 2),(1, 1, 2)]`, X = `[(x), (y), (z)]` and B = `[(1), (2), (4)]`
Applying R2 → R2 – 2R1 and R3 – R1, we get
`[(1, 1, 1),(0, 1, 0),(0, 0, 1)][(x), (y), (z)] = [(1), (0), (3)]`
Hence, the original matrix A is reduced to an upper triangular matrix.
∴ `[(x + y + z),(0 + y + 0),(0 + 0 + z)] = [(1), (0), (3)]`
∴ By equality of martices, we get
x + y + z = 1 ...(i)
y = 0
z = 3
Substituting y = 0 and z = 3 in equation (i), we get
x + 0 + 3 = 1
∴ x = 1 – 3 = – 2
∴ x = –2, y = 0 and z = 3 is the required solution.
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