Question

If matrix A = `[(-p, q),(r, p)]` is such that A² = I, then ______.

Options

• 1 + p² + qr = 0

• 1 − p² − qr = 0

• 1 − p² + qr = 0

• 1 + p² − qr = 0

Answer 1

If matrix A = `[(-p, q),(r, p)]` is such that A² = I, then 1 − p² − qr = 0.

Explanation:

A² = A × A

= `[(-p, q),(r, p)] [(-p, q),(r, p)]`

= `[((-p)(-p) + (q)(r), (-p)(q) + (q)(p)),((r)(-p) + (p)(r), (r)(q) + (p)(p))]`

= `[(p^2 + qr, -pa + pq),(-rp + pr, qr + p^2)]`

= `[(p^2 + qr, 0),(0, p^2 + qr)]`

Equate A² to the Identity Matrix (I):

A² = I

I = `[(1, 0),(0, 1)]`

`[(p^2 + qr, 0),(0, p^2 + qr)] = [(1, 0),(0, 1)]`

Compare the elements:

p² + qr = 1

1 − p² − qr = 0