Let A = [ a, b; c, d ] and B = [ α; β ] ≠ [ 0; 0 ] such that AB = B and a + d = 2021, then the value of ad - bc is equal to ___________.

Solution

Maths sir · 4 steps

AB = B with B ≠ 0 forces det(A - I) = 0. Expand that and the trace does the rest.

1. Move B across. Write AB = B as AB - IB = O, that is (A - I)B = O. Here B is a nonzero column, so this homogeneous system has a non-trivial solution.
2. Singular matrix. A homogeneous system has a non-trivial solution only when its coefficient matrix is singular. So det(A - I) = 0.

   |a-1, b; c, d-1| = (a-1)(d-1) - bc = 0

3. Expand. Expanding gives ad - a - d + 1 - bc = 0, so ad - bc = (a + d) - 1.
4. So. With a + d = 2021, ad - bc = 2021 - 1 = 2020.

The idea to keep. Whenever AX = kX for some nonzero X, rewrite it as (A - kI)X = O and set det(A - kI) = 0. For a 2 × 2 matrix that links trace and determinant in one line.

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