Modular Determinant of a Matrix

The modular determinant of a matrix is its determinant computed modulo a given number n.

To find the determinant modulo n, we first calculate the determinant of the matrix as usual.

Then, we take the result and apply the modulo n operation to obtain the modular determinant.

What is a modulus? The modulus of a number is the remainder left after dividing it by another number, known as the divisor. In modular arithmetic, this operation is written as a mod  n and tells us what’s left over when a is divided by n. For example, $$ 8 \mod 5 = 3 $$ because when 8 is divided by 5, the remainder is 3. This concept is widely used in cryptography.

A Practical Example

Let’s compute the modular determinant of a 2×2 matrix modulo 5.

$$ A = \begin{pmatrix} 8 & -1 \\ 3 & 2 \end{pmatrix} \ \mod \ 5 $$

First, we calculate its determinant:

$$ \det(A) \ \mod \ 5 = ( 8 \cdot 2 ) - ( -1 \cdot 3 ) \ \mod \ 5 $$

$$ \det(A) \ \mod \ 5= 16 + 3 \ \mod \ 5 $$

$$ \det(A) \ \mod \ 5= 19 \ \mod \ 5 $$

Now, we apply the modulo 5 operation.

Since 19 divided by 5 leaves a remainder of 4, we get:

$$ \det(A) \ \mod \ 5 = 4 $$

Note. The modular determinant depends on the chosen modulus. Changing the modulus can significantly alter the result, even for the same matrix.

Another Example

Now, let’s compute the modular determinant of the same matrix using modulus 6.

$$ A = \begin{pmatrix} 8 & -1 \\ 3 & 2 \end{pmatrix} \ \mod \ 6 $$

We follow the same steps as before:

$$ \det(A) \ \mod \ 6 = ( 8 \cdot 2 ) - ( -1 \cdot 3 ) \ \mod \ 6 $$

$$ \det(A) \ \mod \ 6= 16 + 3 \ \mod \ 6 $$

$$ \det(A) \ \mod \ 6= 19 \ \mod \ 6 $$

Now, applying the modulo 6 operation:

Since 19 divided by 6 leaves a remainder of 1, we get:

$$ \det(A) \ \mod \ 6 = 1 $$

Notice how the determinant is different when computed modulo 5 versus modulo 6.

And so on.