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Question
Solve the following equations for X and Y, if 3X − Y = `[(1, -1),(-1, 1)]` and X – 3Y = `[(0, -1),(0, -1)]`.
Solution
3X – Y = `[(1, -1),(-1, 1)]` ...(1)
X – 3Y = `[(0, -1),(0, -1)]` ...(2)
Multiplying (1) by 3, we get,
9X – 3Y = `[(3, -3),(-3, 3)]`
Subtracting (2) from this equation, we get,
8X = `[(3, -3),(-3, 3)] - [(0, -1),(0, -1)]`
= `[(3 - 0, -3-(-1)),(-3 - 0, 3 - ( - 1))]`
= `[(3, -2),(-3, 4)]`
∴ X = `1/8[(3, -2),(-3, 4)]`
= `[(3/8, (-1)/4),((-3)/8, 1/2)]`
From (1), Y = `3"X" - [(1, -1),(-1, 1)]`
= `3[(3/8, (-1)/4),((-3)/8, 1/2)] - [(1, -1),(-1, 1)]`
= `[(9/8, (-3)/4),((-9)/8, 3/2)] - [(1, -1),(-1, 1)]`
= `[(9/8 - 1, (-3)/4 - (-1)),((-9)/8 - (-1), 3/2 - 1)]`
= `[(1/8, 1/4),((-1)/8, 1/2)]`
∴ X = `[(3/8, (-1)/4),((-3)/8, 1/2)] and "Y" = [(1/8, 1/4),((-1)/8, 1/2)]`.
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