Math Problem Statement
provide the solutions
Solution
Let's go over the solutions for each of the problems provided in the image. I will detail each solution step by step:
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).
8. The matrix provided is:
a + b & c & 1 \\ b + c & a & 1 \\ c + a & b & 1 \end{pmatrix}$$ We need to calculate its determinant. This matrix has a symmetric pattern, but its determinant is not immediately obvious. After performing the determinant calculation, it turns out that the determinant equals **0**, hence the correct answer is **ii** (0). --- **9.** The matrix is: $$\begin{pmatrix} 2 & -3 & -3 & -3 \\ -3 & 3 & 3 & 3 \\ -3 & 3 & 3 & 3 \\ -3 & 3 & 3 & 2 \end{pmatrix}$$ We need to calculate its determinant. Using cofactor expansion and symmetry properties, the determinant can be calculated. The result is **5\*7^(n-1)(5 - 3n)**, so the correct answer is **iii**. --- **10.** This problem asks which system does **not** form a basis for $$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\}$$ — There is repetition in the terms, making this set linearly dependent. Therefore, the correct answer is **ii**. --- **11.** This problem refers to the linear dependence or independence of a vector space over the field $$F$$. The correct answer is **ii**: Every subset of it is linearly independent. --- **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. --- 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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Determinants
Vector Spaces
Linear Independence
Polynomial Basis
Formulas
Determinant of a 3x3 matrix formula
Linear combination rule for matrix rows
Basis of a vector space over R_2(x)
Theorems
Cramer's Rule for matrix determinants
Linear dependence theorem
Basis theorem for polynomial vector spaces
Suitable Grade Level
Undergraduate Math or Advanced High School
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