Determinants Matrices Question 7
Question: If $ A= \begin{bmatrix} a & b \\ 0 & a \\ \end{bmatrix} $ is nth root of $ I_2 $ , then choose the correct statements:
(i) if n is odd, $ a=1,b=0 $
(ii) in n is odd, $ a=-1,b=0 $
(iii) if n is even, $ a=1,b=0 $
(iv) if n is even, $ a=-1,b=0 $
Options:
A) i, ii, iii
B) ii, iii, iv
C) i, ii, iii, iv
D) i, iii, iv
Show Answer
Answer:
Correct Answer: D
Solution:
- [d] if A is nth root of $ I_2 $ , then $ A^{n}=I_2 $ . Now, $ A^{2}= \begin{bmatrix} a & b \\ 0 & a \\ \end{bmatrix} \begin{bmatrix} a & b \\ 0 & a \\ \end{bmatrix} = \begin{bmatrix} a^{2} & 2ab \\ 0 & a^{2} \\ \end{bmatrix} $
$ A^{3}=A^{2}A= \begin{bmatrix} a^{2} & 2ab \\ 0 & a^{2} \\ \end{bmatrix} \begin{bmatrix} a & b \\ 0 & a \\ \end{bmatrix} = \begin{bmatrix} a^{3} & 3a^{2}b \\ 0 & a^{3} \\ \end{bmatrix} $ Thus, $ A^{n}= \begin{bmatrix} a^{n} & n{a^{n-1}}b \\ 0 & a^{n} \\ \end{bmatrix} $ Now, $ A^{n}=I\Rightarrow \begin{bmatrix} a^{n} & n{a^{n-1}}b \\ 0 & a^{n} \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $
$ \Rightarrow a^{n}=1,b=0 $
BETA
