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Q.
If $A^{2}=I,$ then the value of ${det}(A-I)$ is (where $A$ has order 3)
Matrices
Solution:
det (A-I)=det $\left(A-A^{2}\right)$ =det $A(I-A)$ =det A $\cdot$ det $(I-A) =- $det $A \cdot$ det $(A-I)$
Now $A^{2}=I$
$\Rightarrow $ det $\left(A^{2}\right)=$ det $(I) \Rightarrow ( $ det $A)^{2}=1 \Rightarrow $ det $(A)=\pm 1$
Thus, det $(A)$ can be $1$ or $-1,$ from which we cannot say anything about det $(A-I)$