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How should I approach these sort of problems, like in algebraic problems of matrices, there are several restrictions with it, we cannot divide it by the matrix itself (like matrix $A$ in the given ques) always because $|A|$ might be 0 (which is the case most of the times)

For an example, what is the way to find value of the determinant given in the question?

A a nilpotent matrix with index 2.

(In this problem we have to find the value of $|A+3I|^{50}$, which seems exactly south to the info given). The problem is:

If $A$ is a $3\times3$ Nilpotent matrix such that $|25A+3I|=\frac1{3^{23}}$, then $\det((A+3I)^{50})$ is...

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    $\begingroup$ Please use MathJax instead of including images in posts (images cannot be found by searches). $\endgroup$
    – Greg Martin
    Mar 18 at 17:16
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    $\begingroup$ I will not say that it's a twisted mind who has conceived such an exercise, but I would never have given such a bizarre exercise to my students... $\endgroup$
    – Jean Marie
    Mar 18 at 17:20
  • $\begingroup$ use the fact that $A$ is similar to a strictly upper triangular matrix and similarity transforms don't change determinants $\endgroup$
    – user8675309
    Mar 18 at 17:45
  • $\begingroup$ @DietrichBurde please tell the answer which you are getting by your method. $\endgroup$
    – Akshat Saxena
    Mar 18 at 18:13
  • $\begingroup$ @JeanMarie we have to solve these kinda problems to prepare for an exam in India. $\endgroup$
    – Akshat Saxena
    Mar 18 at 18:14

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