The document contains multiple choice questions focused on matrix operations, determinants, and linear algebra concepts. Here's how to solve a few representative tasks from the file:

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Task Example: Determinant Calculation

Task (16):
Given $A = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$, find $\det(A)$.

Solution: $\det(A) = (5)(8) - (6)(7) = 40 - 42 = -2$ Answer: a. $-2$

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Task Example: Matrix Multiplication

Task (1):
Multiply $\begin{pmatrix} 2 & 1 \\ 3 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$.

Solution: Perform standard matrix multiplication:

\begin{pmatrix} (2)(1)+(1)(1) & (2)(-1)+(1)(1) \\ (3)(1)+(1)(1) & (3)(-1)+(1)(1) \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ 4 & -2 \end{pmatrix}$$ **Answer:** a. $$\begin{pmatrix} 3 & -1 \\ 4 & -2 \end{pmatrix}$$ --- ### Task Example: System of Linear Equations **Task (26):** Solve the system of equations: $$2x + y = 4 \\ x - 3y = -5$$ **Solution:** 1. Solve for $$x$$ from the first equation: $$x = 2 - \frac{y}{2}$$ 2. Substitute into the second equation: $$2 - \frac{y}{2} - 3y = -5 \\ 2 - \frac{7y}{2} = -5 \\ \frac{7y}{2} = 7 \\ y = 2$$ 3. Substitute $$y = 2$$ back into the first equation: $$2x + 2 = 4 \\ x = 1$$ **Answer:** a. (1, 2) --- If you'd like me to work on a specific problem from the document, let me know the question number, and I’ll provide the solution. Here are **five related questions** for further exploration: 1. How do you find the inverse of a $$2 \times 2$$ matrix? 2. What is the geometric interpretation of a determinant in 2D? 3. How do you test if a system of equations has no solutions? 4. How does matrix multiplication differ from scalar multiplication? 5. How do eigenvalues relate to the determinant? **Tip:** When calculating determinants, always verify if properties like row/column swaps or zeros can simplify the calculation.