The problem provided is in Estonian, and I will first translate it to make sure the question is clear.

Translation:

1. Let $A$ be a matrix of order 4, and its determinant is given as $|A| = -2$.
2. Let matrix $B$ be the inverse of matrix $A$, and matrix $C$ be the opposite (negative) of matrix $A$.
3. Find the following determinants:
   • $|C|$
   • $|AB|$
   • $|A + C|$
   • $|CB|$

Now, let's proceed with solving these step by step:

1. $|C|$

Matrix $C$ is the opposite of matrix $A$, so: $C = -A$ For determinants, we know that: $|C| = |-A| = (-1)^n \cdot |A|$ Where $n$ is the order of the matrix. Since $A$ is a 4x4 matrix ($n = 4$): $|C| = (-1)^4 \cdot |A| = 1 \cdot (-2) = -2$ Thus, $|C| = -2$.

2. $|AB|$

Matrix $B$ is the inverse of matrix $A$, so: $AB = I \quad \text{(the identity matrix)}$ The determinant of the identity matrix is always 1, regardless of the size: $|AB| = |I| = 1$

3. $|A + C|$

Since $C = -A$, we have: $A + C = A - A = 0 \quad \text{(the zero matrix)}$ The determinant of the zero matrix is 0: $|A + C| = 0$

4. $|CB|$

Matrix $C = -A$, so: $CB = (-A)B = -(AB)$ We already know that $AB = I$, so: $CB = -I$ The determinant of $-I$ is $(-1)^n \cdot |I|$. Since $n = 4$: $|CB| = (-1)^4 \cdot |I| = 1 \cdot 1 = 1$

Final answers:

• $|C| = -2$
• $|AB| = 1$
• $|A + C| = 0$
• $|CB| = 1$

Do you have any questions or want further details on any step?

Here are five related questions to expand on the topic:

1. What properties of determinants are used to calculate $|C|$ from $A$?
2. How does the determinant of the identity matrix always remain 1?
3. Why is the determinant of a sum involving opposites of matrices zero?
4. How would the results change if $A$ were a 3x3 matrix instead?
5. Can a zero matrix ever have a non-zero determinant?

Tip: The determinant of a matrix's inverse is always the reciprocal of the original matrix's determinant.