Short Answer

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

Therefore, the $d e t A$ indeed must be divisible by 8

So, the given statement is true.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 “ $m \times n$ .”

Step 2: 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 $d e t B$ 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

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Matrix Definition

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

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Linear Algebra With Applications

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Short Answer

If A is a 4×4 matrix whose entries are all 1 or -1 , then detAmust be divisible by 8 (i.e., detA=8k for some integer k ).

Step by Step Solution

Step 1: Matrix Definition

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

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