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Question
Solve the following by inversion method 2x + y = 5, 3x + 5y = −3
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Solution
Matrix form of the given system of equations is
`[(2, 1),(3, 5)] [(x), (y)] = [(5),(-3)]`
This is of the form AX = B,
where A = `[(2, 1),(3, 5)]`, X = `[(x),(y)]` and B = `[(5),(-3)]`
To determine X, we have to find A−1
|A| = `[(2, 1),(3, 5)]`
= 10 − 3
= 7 ≠ 0
∴ A−1 exists.
Consider AA−1 = I
∴ `[(2, 1),(3, 5)]` A−1 = `[(1, 0),(0, 1)]`
Applying R2 → 2R2 − 3R1, we get
`[(2, 1),(0, 7)]` A−1 = `[(1, 0),(-3, 2)]`
Applying R1 → 7R1 – R2, we get
`[(14, 0),(0, 7)]` A−1 = `[(10, -2),(-3, 2)]`
Applying R1 → `(1/14)` R1 and R2 → `(1/7)` R2, we get
`[(1, 0),(0, 1)]` A−1 = `[(10/14, (-2)/14),((-3)/7, 2/7)]`
∴ A−1 = `1/7[(5, -1),(-3, 2)]`
Pre-multiplying AX = B by A−1, we get
A−1(AX) = A−1B
∴ (A−1A)X = A−1B
∴ IX = A−1B
∴ X = A−1B
∴ X = `1/7[(5, -1),(-3, 2)] [(5),(-3)]`
∴ `[(x),(y)] = 1/7[(25 + 3),(-15 - 6)]`
= `1/7[(28),(-21)]`
= `[(4),(-3)]`
∴ By equality of matrices, we get
x = 4, y = – 3
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