Question

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)

Answer 1

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)}`