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Question
Solve the system of linear equations using the matrix method.
2x – y = –2
3x + 4y = 3
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Solution
Given system of equations,
2x – y = –2
3x + 4y = 3
The system of equations can be written as AX = B. Hence, X = A−1B.
A = `[(2,-1),(3,4)]`, X = `[(x),(y)]` and B = `[(-2),(3)]`
⇒ |A| = `|(2,-1),(3,4)|`
= 8 + 3
= 11 ≠ 0
The cofactors of the elements of matrix A are as follows:
A11 = (−1)1+1 (4) = 4
A12 = (−1)1+2 (3) = −3
A21 = (−1)2+1 (−1) = 1
A22 = (−1)2+2 (2) = 2
Matrix made of elements of cofactor of A = `[(4,-3),(1,2)]`
adj A = `[(4,-3),(1,2)] = [(4,1),(-3,2)]`
A−1 = `1/|A|` (adj A)
= `1/11 [(4,1),(-3,2)]`
∴ X = A−1B
= `1/11 [(4,1),(-3,2)] [(-2),(3)]`
= `1/11 [(-8 + 3),(6 + 6)]`
= `1/11 [(-5),(12)]`
= `[(-5/11),(12/11)]`
⇒ `[(x),(y)] = [(-5/11),(12/11)]`
⇒ x = `-5/11` and y = `12/11`
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