Q5. For A with rows 1, 2 and 4, 2, show that |2A| = 4|A| (NCERT Exercise 4.1, Q3).
|A| = 2 − 8 = −6. 2A has rows 2, 4 and 8, 4, so |2A| = 8 − 32 = −24 = 4 × (−6) = 4|A|.
Q6. Evaluate the determinant with rows a, b, c; b, c, a; c, a, b when a = 1, b = 2, c = 3, and check that it equals −(a3 + b3 + c3 − 3abc).
Along row 1: 1(3 × 2 − 1 × 1) − 2(2 × 2 − 1 × 3) + 3(2 × 1 − 3 × 3) = 5 − 2 − 21 = −18. Also a3 + b3 + c3 − 3abc = 1 + 8 + 27 − 18 = 18, and the determinant is its negative.
Q7. A is a 3 × 3 matrix with |A| = 5. Find the determinants of −2A and of A′.
The determinant of −2A is (−2)3 × 5 = −40, and |A′| = |A| = 5 (a property of determinants, beyond the NCERT text).
Q8. Find the values of x for which the determinant with rows x, 3 and 3, x is 0, and those for which the determinant with rows 1, x, x2; 1, 2, 4; 1, 3, 9 is 0.
x2 − 9 = 0 gives x = ±3. The 3 × 3 determinant is (2 − x)(3 − x)(3 − 2) after expanding along row 1: 1(18 − 12) − x(9 − 4) + x2(3 − 2) = x2 − 5x + 6 = (x − 2)(x − 3). It is 0 for x = 2 or 3, when two rows are equal.