If the matrices A = <table><tr><td>1</td><td>1</td><td>2</td></tr><tr><td>1</td><td>3</td><td>4</td></tr><tr><td>1</td><td>-1</td><td>3</td></tr></table>, B = adj A and C = 3A, then <sup>|adj B|</sup>/<sub>|C|</sub> is equal to :
Step-by-Step Solution
Key Concept: Use properties of determinants: |adj A| = |A|^(n-1), |adj B| = |B|^(n-1), and |kA| = k^n |A| for a 3x3 matrix.
Given A = [[1, 1, 2], [1, 3, 4], [1, -1, 3]]. |A| = 1(9+4) - 1(3-4) + 2(-1-3) = 13 + 1 - 8 = 6. Since B = adj A, |B| = |adj A| = |A|^(3-1) = |A|^2 = 6^2 = 36. Then |adj B| = |B|^(3-1) = |B|^2 = 36^2 = 1296. Given C = 3A, |C| = |3A| = 3^3 |A| = 27 * 6 = 162. Therefore, |adj B| / |C| = 1296 / 162 = 8.
Correct Answer: 3