Let A be a symmetric matrix such that |A|=2 and [ 2, 1; 3, 3/2 ]A = [ 1, 2; α, β ]. If the sum of the diagonal elements of A is s, then (β s)/α² is equal to __________.

Solution

Maths sir · 4 steps

Look at the left matrix: row 2 is 3/2 times row 1, so the product's row 2 is too. That gives α and β at once.

1. Rows carry over. In (2, 1; 3, 3/2), the second row is 3/2 times the first. Row i of a product is row i of the left matrix times A, so row 2 of the result is 3/2 times (1, 2). That gives α = 3/2 and β = 3.
2. Set up A. A symmetric 2 × 2 matrix is A = (a, b; b, c). Row 1 of the product gives 2a + b = 1 and 2b + c = 2, so b = 1 - 2a and c = 4a.

   |A| = ac - b² = 4a² - (1-2a)² = 4a - 1 = 2

3. Find the trace. So a = 3/4, c = 3 and b = -1/2. The sum of the diagonal elements is s = 3/4 + 3 = 15/4.
4. So. (β s)/α² = (3 × 15/4)/(9/4) = 45/9 = 5.

The idea to keep. Row i of PA is row i of P times A. If two rows of P are proportional, the same rows of PA are proportional in the same ratio. Spot this before you multiply anything out.

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