Question

If A = `[(3, -4),(-1, 2)]`, find matrix B such that BA = I,where I is unity matrix of order 2

Answer 1

A = `[(3, -4),(-1, 2)]`
BA = I, where I is unity matrix of order 2

∴ I = `[(1, 0),(0, 1)]`

Let B = `[(a, b),(c, d)]`

∴ BA = `[(a, b),(c,d)] xx [(3, -4),(-1, 2)] `

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

∴ `[(3a - b, -4a + 2b),(3c - d, -4c + 2d)] = [(1, 0),(0, 1)]`
Comparing the corresponding terms, we get
3a – b = 1,
–4a + 2b = 0
⇒2b = 4a
⇒ b = 2a
∴ 3a – b = 1
⇒  3a – 2a = 1
⇒ a = 1
and
b = 2a
⇒ b = 2 x 1 = 2
∴ a = 1, b = 2
and
3c – d = 0
⇒ d = 3c
–4c + 2d = 1
⇒ –4c + 2 x 3c = 1
⇒ –4c + 6c = 1
⇒ 2x = 1
⇒ c = `(1)/(2)`
and
d = 3c = `3 xx (1)/(2) = (3)/(2)`

Hence a = 1, b = 2, c = `(1)/(2), "d" = (3)/(2)`

∴ Matrix B = `[(1, 2),(1/2, 3/2)]`.