Question

Solve the following equations for X and Y, if 3X − Y = `[(1, -1),(-1, 1)]`  and X – 3Y = `[(0, -1),(0, -1)]`.

Answer 1

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)]`.