Question:
$\begin{bmatrix}0&a\\ b&0\end{bmatrix}^{^4}=I$, then
$\begin{bmatrix}0&a\\ b&0\end{bmatrix}^{^4}=I$, then
Updated On: Apr 28, 2024
- $a = 1 = 2b$
- $a = b$
- $a = b^2$
- $ab = 1$
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The Correct Option is D
Solution and Explanation
Here,$\begin{bmatrix}0&a\\ b&0\end{bmatrix}^{^2}=\begin{bmatrix}0&a\\ b&0\end{bmatrix}\begin{bmatrix}0&a\\ b&0\end{bmatrix}^{^2}$
$=\begin{bmatrix}0&+&ab&0&+&0\\ 0&+&0&ab&+&0\end{bmatrix}=\begin{bmatrix}ab&0\\ 0&ab\end{bmatrix}$
Similarly,$\begin{bmatrix}0&a\\ b&0\end{bmatrix}^{^4}=\begin{bmatrix}ab&0\\ 0&ab\end{bmatrix}\begin{bmatrix}ab&0\\ 0&ab\end{bmatrix}$
$=\begin{bmatrix}a^{2}&b^{2}&+&0&0&+&0&\\ 0&+&0&&0&+&a^{2}&b^{2}\end{bmatrix}=\begin{bmatrix}a^{2}&b^{2}&0&\\ 0&&a^{2}&b^{2}\end{bmatrix}$
Now,
$\begin{bmatrix}0&a\\ b&0\end{bmatrix}^{^4}=I \Rightarrow \begin{bmatrix}a^{2}&b^{2}&0&\\ 0&&a^{2}&b^{2}\end{bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}$
$\Rightarrow a^{2}b^{2}=1\Rightarrow ab=1$
$=\begin{bmatrix}0&+&ab&0&+&0\\ 0&+&0&ab&+&0\end{bmatrix}=\begin{bmatrix}ab&0\\ 0&ab\end{bmatrix}$
Similarly,$\begin{bmatrix}0&a\\ b&0\end{bmatrix}^{^4}=\begin{bmatrix}ab&0\\ 0&ab\end{bmatrix}\begin{bmatrix}ab&0\\ 0&ab\end{bmatrix}$
$=\begin{bmatrix}a^{2}&b^{2}&+&0&0&+&0&\\ 0&+&0&&0&+&a^{2}&b^{2}\end{bmatrix}=\begin{bmatrix}a^{2}&b^{2}&0&\\ 0&&a^{2}&b^{2}\end{bmatrix}$
Now,
$\begin{bmatrix}0&a\\ b&0\end{bmatrix}^{^4}=I \Rightarrow \begin{bmatrix}a^{2}&b^{2}&0&\\ 0&&a^{2}&b^{2}\end{bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}$
$\Rightarrow a^{2}b^{2}=1\Rightarrow ab=1$
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Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
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