If A = ( 0, sin α; sin α, 0 ) and det ( A² - 1/2I ) = 0, then a possible value of α is :

Solution

Maths sir · 3 steps

Square A first: it collapses to sin² α times the identity, and the rest is one line.

1. Square the matrix. Multiplying row by column, each diagonal entry of A² is sin² α and each off-diagonal entry is 0. So A² = sin² α I.
2. Subtract half of I. A² - 1/2I = (sin² α - 1/2) I, a diagonal matrix with the same number twice on the diagonal.

   det(A² - 1/2I) = (sin² α - 1/2)² = 0 ⇒ sin² α = 1/2

3. So. sin α = ±1/√2, and among the options only α = π/4 works. The answer is (a).

The idea to keep. When a matrix squares to a multiple of I, write A² = kI first. Then det(A² - cI) = (k - c)ⁿ for an n × n matrix, and no long expansion is needed.

Answer from the NTA final answer key. Solution by PrepWiser.