Mathematics: Let [11 0 0 13 4] 12 5 0 17 19 b= | 0 0 0 1 0 21 23 3 29 37 | 2 0 0 35 0 compute det(b100)

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Let [11 0 0 13 4] 12 5 0 17 19 b= | 0 0 0 1 0 21 - id1100134125017190001021231851470, beverly9662

hoyanna69

31.12.2021

Looks like

$B=\begin{bmatrix}11&0&0&13&4\\12&5&0&17&19\\0&0&0&1&0\\21&23&3&29&37\\2&0&0&35&0\end{bmatrix}$

Recall that for any two matrices A,B ,

$\det(AB)=\det A\det B$

which means

$\det(B^{100})=(\det B)^{100}$

Laplace expansion along the third row gives us

$\det B=-\det\begin{bmatrix}11&0&0&4\\12&5&0&19\\21&23&3&37\\2&0&0&0\end{bmatrix}$

Expansion along the last row gives

$\det B=-\left(-2\det\begin{bmatrix}0&0&4\\5&0&19\\23&3&37\end{bmatrix}\right)=2\det\begin{bmatrix}0&0&4\\5&0&19\\23&3&37\end{bmatrix}$

Expansion along the first row gives

$\det B=2\left(4\det\begin{bmatrix}5&0\\23&3\end{bmatrix}\right)=8\det\begin{bmatrix}5&0\\23&3\end{bmatrix}$

and finally

$\det\begin{bmatrix}5&0\\23&3\end{bmatrix}=15$

so that

$\det B=8\cdot15=120$

Then

$\boxed{\det(B^{100})=120^{100}}$

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