Question

Help with matrices

Answer 1

At what value of x will the determinant of this matrix equal 0 in mod 11?
\[\left[\begin{matrix}6 & x \\ 4 & 5\end{matrix}\right]\]

Answer 2

30 - 4x = 0 (mod 11)
x = 7.5
What do I do now?

Answer 3

30-4x=0
30=4x
30/4=x
x=7.5

7.5mod(11) ≡ r
\[\frac{ 7.5 }{ 11 } ≡ r\]

r is the remainder

\[\frac{ 7.5 }{ 11 } = 0 r7.5\]

where 0 means 11 goes into 7.5 0 times, and leaves a remainder 7.5

so your answer is just 7.5

Answer 4

The determinant is \(30 - 4x\) indeed.

But after that, you want more solutions. Recall: \(X = 0 (\mod 11)\) means that \(X=11n\) for some \(n\in\mathbb{Z}\).

So, \(30-4x=0 (\mod 11)\) means that \(30-4x=11n \iff (30-11n)=4x
\\\iff x = \frac{30-11n}4 \).

For any value \(n\in\mathbb{Z}\) that you will choose, the resulting \(x\) will verify the described property (determinant=0 (mod 11)).