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

Given equations are

3X – Y = `[(1, -1),(-1, 1)]`          ...(i)

and X – 3Y = `[(0, -1),(0, -1)]`     ...(ii)

By (i) x 3 – (ii), we get

8X = `3[(1, -1),(-1, 1)] - [(0, -1),(0, -1)]`

= `[(3, -3),(-3, 3)] - [(0, -1),(0, -1)]`

= `[(3 - 0, -3 + 1),(-3 - 0, 3 + 1)]`

∴ 8X = `[(3, -2),(-3, 4)]`

∴ X = `(1)/(8) [(3, -2),(-3, 4)]`

∴ X = `[(3/8, (-2)/8),((-3)/8 , 4/8)]`

= `[(3/8 , (-1)/4),((-3)/8, 1/2)]`

By (i) – (ii) x 3, we get

8Y = `[(1, -1),(-1, 1)] - 3[(0, -1),(0, -1)]`

= `[(1, -1),(-1, 1)] - [(0, -3),(0, -3)]`

= `[(1 - 0, -1 + 3),(-1 - 0, 1 + 3)]`

∴ 8Y = `[(1, 2),(-1, 4)]`

∴ Y = `(1)/(8)[(1, 2),(-1, 4)]`

= `[(1/8, 2/8),((-1)/8, 4/8)]`

= `[(1/8, 1/4),((-1)/8, 1/2)]`.