Question

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If A= `[(1, 3),(2, 1),(3, -1)], B = [(2, 1),(1, 2),(1, 1)]` and A + B + C = 0, then C = `[(-3, -4),(-3, -3),(-4, 0)]`
2. If A = `[(3, 1),(-1, 2)]`, then A² = `[(9, 1),(1, 4)]`

Options

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Only (i)

Explanation:

(i)

A = `[(1, 3),(2, 1),(3, -1)]`

B = `[(2, 1),(1, 2),(1, 1)]`

A + B + C = 0

C = −(A + B)

A + B = `[(1, 3),(2, 1),(3, -1)] + [(2, 1),(1, 2),(1, 1)]`

= `[(1 + 2, 3 + 1),(2 + 1, 1 + 2),(3 + 1, -1 + 1)]`

= `[(3, 4),(3, 3),(4, 0)]`

C = −(A + B)

= `- [(3, 4), (3, 3),(4, 0)]`

= `[(-3, -4),(-3, -3),(-4, 0)]`

Therefore, statement (i) is valid.

(ii)

A = `[(3, 1),(-1, 2)]`

A² = `[(3, 1),(-1, 2)] [(3, 1),(-1, 2)]`

= `[(3 xx 3 + 1 xx (-1), 3 xx 1 + 1 xx 2),(-1 xx 3 + 2 xx (-1), -1 xx 1 + 2 xx 2)]`

= `[(9 - 1, 3 + 2),(-3 - 2, -1 + 4)]`

= `[(8, 5),(-5, 3)]`

Therefore, statement (ii) is invalid.