Question

IfA 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).

Short answer

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

So, the given statement is true.

Step-by-step solution

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 “ $m\times n$.”

To check whether the given condition is true or false

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, $\det B$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 detA indeed must be divisible by 8