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Q44E

Expert-verified
Found in: Page 309

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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

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

So, the given statement is true.

See the step by step solution

Step 1: Matrix Definition

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “m by n ” matrix, written “ $m×n$.”

Step 2: To check whether the given condition is true or false

If C is a $2×2$ 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 $3×3$ 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 $4×4$ matrix A , also containing this very same property, using several other properties of the determinant, we conclude that detA indeed must be divisible by 8