If [(2, 3)(0, -1)], then the value of det (A^4) + det (A^10 - (Adj(2A))^10) is equal to ________.

Question

If \(\begin{bmatrix} 2 & 3\\ 0 & -1\\ \end{bmatrix},\) then the value of det (A⁴) + det (A¹⁰ - (Adj(2A))¹⁰) is equal to ________.

Answer 1

Answer is (16)

Now, characteristic roots of A are 2 and –1.

So, characteristic roots of A²⁰ are 2¹⁰ and 1.

Hence, (A²⁰ – 2¹⁰ I) (A²⁰ – I) = 0

\(\Rightarrow\) \(|A^{20} - 2^{10}I| = 0\) (as A²⁰ ≠ I)

\(\Rightarrow\) E = 16