To determine whether the vectors $\mathbf{x}_1 = [q, 2, 3]$, $\mathbf{x}_2 = [3, -2, 1]$, and $\mathbf{x}_3 = [1, -6, -5]$ are linearly independent or dependent, we can form a matrix using these vectors as columns, and then find the determinant of this matrix. If the determinant is zero, the vectors are linearly dependent; otherwise, they are linearly independent.

Let's form the matrix $A$:

q & 3 & 1 \\ 2 & -2 & -6 \\ 3 & 1 & -5 \end{bmatrix}$$ ### Step 1: Find the determinant of the matrix $$A$$. The determinant of a 3x3 matrix is given by: $$\text{det}(A) = q \begin{vmatrix} -2 & -6 \\ 1 & -5 \end{vmatrix} - 3 \begin{vmatrix} 2 & -6 \\ 3 & -5 \end{vmatrix} + 1 \begin{vmatrix} 2 & -2 \\ 3 & 1 \end{vmatrix}$$ ### Step 2: Evaluate the 2x2 determinants. 1. $$\begin{vmatrix} -2 & -6 \\ 1 & -5 \end{vmatrix} = (-2)(-5) - (-6)(1) = 10 + 6 = 16$$ 2. $$\begin{vmatrix} 2 & -6 \\ 3 & -5 \end{vmatrix} = (2)(-5) - (-6)(3) = -10 + 18 = 8$$ 3. $$\begin{vmatrix} 2 & -2 \\ 3 & 1 \end{vmatrix} = (2)(1) - (-2)(3) = 2 + 6 = 8$$ ### Step 3: Plug the values back into the determinant formula. $$\text{det}(A) = q(16) - 3(8) + 1(8)$$ $$\text{det}(A) = 16q - 24 + 8$$ $$\text{det}(A) = 16q - 16$$ ### Step 4: Determine the condition for linear dependence. - The vectors are linearly **dependent** if the determinant is zero: $$16q - 16 = 0$$ $$16q = 16$$ $$q = 1$$ Thus, if $$q = 1$$, the vectors are linearly dependent. - If $$q \neq 1$$, the vectors are linearly **independent**. ### Summary: - The vectors $$\mathbf{x}_1$$, $$\mathbf{x}_2$$, and $$\mathbf{x}_3$$ are linearly dependent when $$q = 1$$. - The vectors are linearly independent when $$q \neq 1$$. --- Do you want details on how to test the condition further or any clarification on this? Here are 5 related questions to consider: 1. What does it mean for vectors to be linearly independent or dependent? 2. How do you find the determinant of larger matrices? 3. Can you visualize linearly independent vectors in 3D space? 4. How would this process change if there were more than three vectors? 5. What is the significance of the determinant in determining independence? **Tip:** When dealing with 3x3 matrices, computing the determinant is a useful way to check if vectors are independent or dependent.