Solveeit Logo
Login

Question

Mathematics Question on Matrices

[0a b0]4=I\begin{bmatrix}0&a\\\ b&0\end{bmatrix}^{^4}=I, then

A

a=1=2ba = 1 = 2b

B

a=ba = b

C

a=b2a = b^2

D

ab=1ab = 1

Answer

ab=1ab = 1

Explanation

Solution

Here,[0a b0]2=[0a b0][0a b0]2\begin{bmatrix}0&a\\\ b&0\end{bmatrix}^{^2}=\begin{bmatrix}0&a\\\ b&0\end{bmatrix}\begin{bmatrix}0&a\\\ b&0\end{bmatrix}^{^2}
=[0+ab0+0 0+0ab+0]=[ab0 0ab]=\begin{bmatrix}0&+&ab&0&+&0\\\ 0&+&0&ab&+&0\end{bmatrix}=\begin{bmatrix}ab&0\\\ 0&ab\end{bmatrix}
Similarly,[0a b0]4=[ab0 0ab][ab0 0ab]\begin{bmatrix}0&a\\\ b&0\end{bmatrix}^{^4}=\begin{bmatrix}ab&0\\\ 0&ab\end{bmatrix}\begin{bmatrix}ab&0\\\ 0&ab\end{bmatrix}
=[a2b2+00+0 0+00+a2b2]=[a2b20 0a2b2]=\begin{bmatrix}a^{2}&b^{2}&+&0&0&+&0&\\\ 0&+&0&&0&+&a^{2}&b^{2}\end{bmatrix}=\begin{bmatrix}a^{2}&b^{2}&0&\\\ 0&&a^{2}&b^{2}\end{bmatrix}
Now,
[0a b0]4=I⇒[a2b20 0a2b2]=[10 01]\begin{bmatrix}0&a\\\ b&0\end{bmatrix}^{^4}=I \Rightarrow \begin{bmatrix}a^{2}&b^{2}&0&\\\ 0&&a^{2}&b^{2}\end{bmatrix}=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}
⇒a2b2=1⇒ab=1\Rightarrow a^{2}b^{2}=1\Rightarrow ab=1