If A=[√2, 1; -1, √2], B=[1, 0; 1, 1], C=ABAᵀ and X=Aᵀ C² A, then det X is equal to :

Solution

Maths sir · 3 steps

Do not multiply a single matrix. Use det(PQ) = det P · det Q and work with numbers only.

1. Small determinants first. det A = (√2)(√2) - (1)(-1) = 2 + 1 = 3. det B = (1)(1) - (0)(1) = 1.
2. Determinant of C. The determinant of a product is the product of the determinants, and det Aᵀ = det A. So det C = det A · det B · det Aᵀ = 3 × 1 × 3 = 9.

   det X = det Aᵀ · (det C)² · det A = 3 × 81 × 3

3. So. det X = 729, which is option (b).

The idea to keep. For a long product of square matrices, never multiply the matrices. Each factor gives its own determinant, a transpose changes nothing, and a power Cⁿ gives (det C)ⁿ.

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