Question

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

Short answer

The determinant of a $3\times 3$ 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 $3\times 3$ 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.

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

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