https://file.aiquickdraw.com/mathbot/file/17269585380833k7da0pw.jpg

Let's go over the solutions for each of the problems provided in the image. I will detail each solution step by step:

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**7.** The problem asks about the determinant of a matrix when one row is a linear combination of the others. If this happens, the determinant of the matrix is always **0**. Thus, the correct answer is **ii** (Eshte i barabartë me 0).

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**8.** The matrix provided is:

\[
\begin{pmatrix}
a + b & c & 1 \\
b + c & a & 1 \\
c + a & b & 1
\end{pmatrix}
\]

We need to calculate its determinant. This matrix has a symmetric pattern, but its determinant is not immediately obvious. After performing the determinant calculation, it turns out that the determinant equals **0**, hence the correct answer is **ii** (0).

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**9.** The matrix is:

\[
\begin{pmatrix}
2 & -3 & -3 & -3 \\
-3 & 3 & 3 & 3 \\
-3 & 3 & 3 & 3 \\
-3 & 3 & 3 & 2
\end{pmatrix}
\]

We need to calculate its determinant. Using cofactor expansion and symmetry properties, the determinant can be calculated. The result is **5\*7^(n-1)(5 - 3n)**, so the correct answer is **iii**.

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**10.** This problem asks which system does **not** form a basis for \( R_2(x) \). 

Let's evaluate each set of polynomials to check if they form a linearly independent set:
- **i**: \(\{x^2, x, x^2 + 2\}\) — This set is linearly dependent since two terms involve \(x^2\).
- **ii**: \(\{-4x, 5x, -2x, -3x\}\) — All terms are multiples of \(x\), so they are linearly dependent.
- **iii**: \(\{2x^2 - 2x + 7, -2x + x^2 + 1\}\) — These polynomials are linearly independent.
- **iv**: \(\{x^2 + 1, x + 1, x^2 + 2x, x^2 + 1\}\) — There is repetition in the terms, making this set linearly dependent.

Therefore, the correct answer is **ii**.

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**11.** This problem refers to the linear dependence or independence of a vector space over the field \(F\). The correct answer is **ii**: Every subset of it is linearly independent.

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**12.** This question refers to conditions under which a square matrix is **not invertible**. A matrix is non-invertible (singular) if its determinant is 0. Thus, the correct answer is **i**: The determinant is equal to 0.

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Would you like further details or have any questions on any specific problem?

Here are five related questions to expand on these problems:
1. What is the process for calculating the determinant of a 3x3 matrix?
2. How can we identify if a set of polynomials is linearly dependent?
3. What properties of a matrix ensure it is invertible?
4. How do linear combinations of matrix rows affect its determinant?
5. What are the steps to perform cofactor expansion for a 4x4 matrix?

**Tip:** Always verify if a matrix is singular by checking its determinant. If it's zero, the matrix cannot be inverted.

Based on the matrix properties and polynomial linear independence questions seen in the image, which involve determinants, linear combinations, and vector spaces.

Explore solutions for matrix determinant calculations and polynomial basis problems, covering topics like linear independence, matrix invertibility, and vector space properties. This guide provides step-by-step solutions, perfect for students tackling advanced algebra and linear algebra concepts.

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