Math Problem Statement
Solution
This problem asks us to determine for which values of the parameter , the given matrix has an inverse. A matrix has an inverse if and only if its determinant is nonzero. The matrix is:
2 & 3 & 1 \\ 0 & a & 2 \\ 2 & -1 & -1 \end{bmatrix}.$$ ### Steps to Solve: 1. Compute the determinant of the matrix, $$\det(A)$$. $$\det(A) = \begin{vmatrix} 2 & 3 & 1 \\ 0 & a & 2 \\ 2 & -1 & -1 \end{vmatrix}.$$ 2. Use the cofactor expansion along the first row: $$\det(A) = 2 \begin{vmatrix} a & 2 \\ -1 & -1 \end{vmatrix} - 3 \begin{vmatrix} 0 & 2 \\ 2 & -1 \end{vmatrix} + 1 \begin{vmatrix} 0 & a \\ 2 & -1 \end{vmatrix}.$$ 3. Compute the $$2 \times 2$$ minors: - $$\begin{vmatrix} a & 2 \\ -1 & -1 \end{vmatrix} = (a)(-1) - (2)(-1) = -a + 2 = 2 - a.$$ - $$\begin{vmatrix} 0 & 2 \\ 2 & -1 \end{vmatrix} = (0)(-1) - (2)(2) = -4.$$ - $$\begin{vmatrix} 0 & a \\ 2 & -1 \end{vmatrix} = (0)(-1) - (a)(2) = -2a.$$ 4. Substitute back into the determinant expression: $$\det(A) = 2(2 - a) - 3(-4) + 1(-2a).$$ 5. Simplify: $$\det(A) = 4 - 2a + 12 - 2a = 16 - 4a.$$ ### Result: The determinant is $$\det(A) = 16 - 4a$$. For the matrix to have an inverse, $$\det(A) \neq 0$$, so: $$16 - 4a \neq 0 \quad \Rightarrow \quad a \neq 4.$$ ### Answer: The correct option is: **c) $$a \neq 4$$.** Would you like a step-by-step explanation for the determinant computation or other details? --- ### Related Questions: 1. What is the general condition for a matrix to have an inverse? 2. How does the determinant of a matrix change if one row is multiplied by a scalar? 3. Can you provide examples of $$2 \times 2$$ matrices with zero determinants? 4. How is the cofactor expansion performed for larger matrices? 5. What is the geometric interpretation of a matrix's determinant? ### Tip: Always check for patterns or shortcuts (like row operations) when calculating large determinants—it can save time!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix algebra
Determinants
Invertible matrices
Formulas
Determinant formula for 3x3 matrices: det(A) = Σ (± a_ij * M_ij), where M_ij are the minors
Condition for invertibility: det(A) ≠ 0
Theorems
Cramer's rule
Matrix determinant property for invertibility
Suitable Grade Level
Grades 11-12 or introductory college-level mathematics
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