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Let $F$ be a field and let $n > 2$ be an integer. Show that $| adj(A)|=|A|^{ n−1}$ for all $A ∈ M_{n×n}(F)$.

Not sure how to do this. Any solutions/hints are greatly appreciated.

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Hint:

$$\DeclareMathOperator{\adj}{adj}\adj A\cdot A= (\det A)I,$$ and $\det$ is a multilinear function of the columns of a matrix. $n$ may be equal to $2$.

Some details:

Since the determinant is multiplicative on matrices, and multilinear on columns (or rows), $$\lvert\adj A \cdot A\rvert=\lvert\adj A\rvert\lvert A\rvert=\lvert A\rvert^n$$ whence $\lvert\adj A\rvert=\lvert A\rvert^{n-1}$ if $\lvert A\rvert\neq 0$.

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  • $\begingroup$ I understand this equality, but I don't see how it can be used just yet. $\endgroup$ – 1233dfv Oct 15 '15 at 21:25
  • $\begingroup$ Just compute the determinants of both sides. The determinant is a multiplicative function of matrices. $\endgroup$ – Bernard Oct 15 '15 at 21:43
  • $\begingroup$ ok, how do I compute the determinant of both sides in this general setting? $\endgroup$ – 1233dfv Oct 15 '15 at 23:46
  • $\begingroup$ I added some details. There remains the case $\det A=0$. $\endgroup$ – Bernard Oct 15 '15 at 23:55
  • $\begingroup$ I see now, thank you. $\endgroup$ – 1233dfv Oct 15 '15 at 23:59

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