If A=pmatrix2&30&-1pmatrix, then the value of (A^4)+[A^10-Adj(2A)^10] is equal to .....

Question

If $A=\begin{pmatrix}2&3\0&-1\end{pmatrix}$, then the value of $\det(A^4)+\det\big[A^{10}-\operatorname{Adj}(2A)^{10}\big]$ is equal to .....

Answer: 16

Accepted Answer

$A^n=\begin{pmatrix}2^n & 2^n-(-1)^n\0 & (-1)^n\end{pmatrix}$ (by induction), so $\det(A^4)=(2^4)(-1)^4=16^{4}$... more directly $\det(A)=-2$, so $\det(A^4)=(-2)^4=16$.

$\operatorname{adj}(2A)=\begin{pmatrix}-2&-6\0&4\end{pmatrix}$, and $[\operatorname{adj}(2A)]^n$ follows a similar triangular pattern; direct computation shows $A^{10}-[\operatorname{adj}(2A)]^{10}=\begin{pmatrix}0 & \ast\0&\ast\end{pmatrix}$, an upper-triangular matrix with a zero diagonal entry in the first row, making its determinant $0$.

So the total is $16+0=16$.

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