Question

If `A = [(-3,2),(2,4)], B = [(1,a),(b,0)]  "and"  (A + B)(A-B) = A^2 - B^2, "Find"  a  "and"  b`

Answer 1

`(A + B)(A-B) = A^2 - B^2`

∴ `A^2 - AB + BA - B^2 = A^2 - B^2`

∴ `-AB + BA = 0`

∴ `AB = BA`

∴`[(-3,2),(2,4)][(1,a),(b,0)] = [(1,a),(b,0)][(-3,2),(2,4)]`

∴`[(-3+2b, -3a+0),(2+4b,2a+0)]=[(-3+2a,2+4a),(-3b+0,2b+0)]`

∴`[(-3+2b,-3a),(2+4b,2a)]=[(-3+2a,2+4a),(-3b, 2b)]`

By equality of matrices

`-3 + 2b = -3 + 2a`   ...(i)

`-3a = 2+4a`       ...(ii)

`2+4b = -3b`    ...(iii)

`2a = 2b`            ...(iv)

From (ii), `7a = -2  ∴ a=-2/7`

From (iii), `7b = -2  ∴ b=-2/7`

These values of `a` and `b` also satisfy equations (i) and (iv).

Hence, `a=-2/7  "and"  b=-2/7`