If A is square matrix of order 3 and \(|A|=6\). If \(|2\,\mathrm{adj}\,(3\,\mathrm{adj}\,(4A))|=2^a3^b\), then \(a+b\) is equal to:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 41
For a matrix B of order 3, the determinant results used are
\(|kB| = k^3|B|\)
\(|\mathrm{adj}\,B| = |B|^{3-1} = |B|^2\)
Working from the inside out:
\(|4A| = 4^3\times 6 = 384 = 2^7\times 3\)
\(|\mathrm{adj}\,(4A)| = (2^7\times 3)^2 = 2^{14}\times 3^2\)
\(|3\,\mathrm{adj}\,(4A)| = 3^3\times 2^{14}\times 3^2 = 2^{14}\times 3^5\)
\(|\mathrm{adj}\,(3\,\mathrm{adj}\,(4A))| = (2^{14}\times 3^5)^2 = 2^{28}\times 3^{10}\)
\(|2\,\mathrm{adj}\,(3\,\mathrm{adj}\,(4A))| = 2^3\times 2^{28}\times 3^{10} = 2^{31}\times 3^{10}\)
So a = 31, b = 10, and
\(a+b = 41\)