1. **Problem:** What is the determinant of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$? 2. **Formula:** The determinant of $A$ is given by $$\det(A) = ad - bc$$. 3. **Explanation:** The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties such as invertibility. 4. **Problem:** Define the minor of an element in a matrix. 5. **Explanation:** The minor of an element is the determinant of the submatrix formed by deleting the row and column of that element. 6. **Problem:** What is the cofactor of an element $a_{ij}$ in a matrix? 7. **Formula:** The cofactor $C_{ij} = (-1)^{i+j} M_{ij}$ where $M_{ij}$ is the minor of $a_{ij}$. 8. **Problem:** How to find the inverse of a matrix $A$? 9. **Formula:** If $\det(A) \neq 0$, then $$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$$ where adj$(A)$ is the adjugate matrix formed by the cofactors. 10. **Problem:** When are two matrices $A$ and $B$ equal? 11. **Explanation:** Two matrices are equal if they have the same dimensions and all corresponding elements are equal. 12. **Problem:** What is a symmetric matrix? 13. **Explanation:** A matrix $A$ is symmetric if $A = A^T$, meaning it equals its transpose. 14. **Problem:** What is a skew-symmetric matrix? 15. **Explanation:** A matrix $A$ is skew-symmetric if $A^T = -A$. 16. **Problem:** What is the result of adding two matrices $A$ and $B$? 17. **Explanation:** The sum $C = A + B$ is a matrix where each element $c_{ij} = a_{ij} + b_{ij}$. 18. **Problem:** How to multiply a matrix $A$ by a scalar $k$? 19. **Explanation:** Each element of $A$ is multiplied by $k$. 20. **Problem:** How to multiply two matrices $A$ and $B$? 21. **Formula:** The element $c_{ij}$ of the product $C = AB$ is $$c_{ij} = \sum_{k} a_{ik} b_{kj}$$. These questions cover determinant, minor, cofactor, inverse, equality, symmetry, skew-symmetry, and matrix operations.