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Q44E

Expert-verifiedFound in: Page 309

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**If** **A**** ** **is a** ${\mathbf{4}}{\mathbf{\times}}{\mathbf{4}}$ **matrix whose entries are all 1** **or -1** **, then ${\mathit{d}}{\mathit{e}}{\mathit{t}}{\mathit{A}}$****must be divisible by 8 (i.e., ${\mathit{d}}{\mathit{e}}{\mathit{t}}{\mathit{A}}{\mathbf{=}}{\mathbf{8}}{\mathit{k}}$**** for some integer k**

Therefore, the $detA$ indeed must be divisible by 8

So, the given statement is true.

If *C* is a $2\times 2$ matrix whose entries are all -1 or 1 , then $detB$ can only be $-2,0$, or 2 .

Using the Laplace expansion along the first row, we conclude that for a $3\times 3$ matrix *B* with the same property, $\mathrm{det}B$ can only be $-6,-4,-2,0,2,4$ , or 6 .

Lastly, also using the Laplace expansion along the first row, we can see that for a $4\times 4$ matrix *A* , also containing this very same property, using several other properties of the determinant, we conclude that det*A* indeed must be divisible by 8

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