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Question:
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Consider the following statements
$1.$ If $\mathrm{A}^{\prime}=\mathrm{A}$; then $A$ is a singular matrix, where $\mathrm{A}^{\prime}$ is the transpose of $A$.
$2.$ If $A$ is a square matrix such that $A^{3}=I$, then $A$ is non-singular. Which of the statements given above is/are correct?
Options:
$1.$ If $\mathrm{A}^{\prime}=\mathrm{A}$; then $A$ is a singular matrix, where $\mathrm{A}^{\prime}$ is the transpose of $A$.
$2.$ If $A$ is a square matrix such that $A^{3}=I$, then $A$ is non-singular. Which of the statements given above is/are correct?
Solution:
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Verified Answer
The correct answer is:
2 only
If $A^{\prime}=\mathrm{A}$ where $A$ ' is transpose of matrix then $|A|=\left|A^{\prime}\right|$
But it is not neccessary that $|A|=0$
i.e. $A$ is singular matrix Hence, statement 1 is wrong. Given $A^{3}=I$
$\left|A^{3}\right|=|I|=1$
$\Rightarrow|A|=1$
Thus, $A$ is non-singular matrix. Hence, only statement 2 is correct.
But it is not neccessary that $|A|=0$
i.e. $A$ is singular matrix Hence, statement 1 is wrong. Given $A^{3}=I$
$\left|A^{3}\right|=|I|=1$
$\Rightarrow|A|=1$
Thus, $A$ is non-singular matrix. Hence, only statement 2 is correct.
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