TS EAMCET · Maths · Matrices
If \(\mathrm{P}\) is a non-singular matrix such that \(\mathrm{I}+\mathrm{P}+\mathrm{P}^2 \ldots+\mathrm{P}^{\mathrm{n}}=\) \(0(0\) denotes the null matrix \()\), then \(\mathrm{P}^{-1}=\)
- A \(\mathrm{P}^{\mathrm{n}}\)
- B \(-\mathrm{P}^{\mathrm{n}}\)
- C \(-\left(1+P+\ldots+P^n\right)\)
- D \(-\left(1+P+\ldots+P^{n-1}\right)\)
Answer & Solution
Correct Answer
(A) \(\mathrm{P}^{\mathrm{n}}\)
Step-by-step Solution
Detailed explanation
\(I+P+P^2+\ldots . .+P^n=0... (i)\) \(\left(I+P+P^2+\ldots . .+P^{n-1}\right)=0-P^n=-P^n\) Pre multiply (i) with \(P^{-1}\) \(\begin{aligned} & P^{-1}+I+P+P^2+\ldots . .+P^{n-1}=0 \\ & P^{-1}=0-\left(I+P+P^2+\ldots .+P^{n-1}\right) \Rightarrow P^{-1}=P^n \end{aligned}\)
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