Q. 62 Prove that the determinant of a ... [FREE SOLUTION]

Chapter 10: Q. 62 (page 825)

Prove that the determinant of a $3 脳 3$ matrix with integer entries is an integer.

Short Answer

Expert verified

The determinant of a $3 脳 3$ matrix with integer entries is an integer because it involves only addition, subtraction and multiplication.

Step by step solution

Step 1. Given Information

Prove that the determinant of a $3 脳 3$ matrix with integer entries is an integer.

Step 2. The determinant of a 3×3 matrix with integer entries is an integer.

It is closed under the operations of addition and multiplication...which means

that all linear combinations of elements in Z yield another element of Z. Thus, since the determinant of a matrix with integer values is a linear combination of integers, it must also be an integer.

Step 3. Let the example A=det123456789

Solving the determinant.

$A = 1 |\begin{matrix} 5 & 6 \\ 8 & 9 \end{matrix}| - 2 |\begin{matrix} 4 & 6 \\ 7 & 9 \end{matrix}| + 3 |\begin{matrix} 4 & 5 \\ 7 & 8 \end{matrix}| A = 1 (45 - 48) - 2 (36 - 42) + 3 (32 - 35) A = 1 (- 3) - 2 (- 8) + 3 (- 3) A = - 3 + 16 - 9 A = 0$

Hence, the determinant of a $3 脳 3$ matrix with integer entries is an integer.

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91影视

Prove that the determinant of a $3 脳 3$ matrix with integer entries is an integer.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The determinant of a 3×3 matrix with integer entries is an integer.

Step 3. Let the example A=det123456789

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91影视

Prove that the determinant of a 3脳3 matrix with integer entries is an integer.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The determinant of a 3×3 matrix with integer entries is an integer.

Step 3. Let the example A=det123456789

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Prove that the determinant of a $3 脳 3$ matrix with integer entries is an integer.