Math Problem Statement
Find all values of k for which the matrix is invertible? A: k=0, B: Any k ≠ 0, C: k=3, D: Any k ≠ 3, E: k = -2, F: Any k ≠ -2, G: Any k ∈ R.
Solution
To determine when the given matrix is invertible, we need to find for which values of the determinant of the matrix is non-zero.
The matrix given is:
2 & 5 & 1 \\ 0 & k & -3 \\ 0 & 3 & 9 \end{pmatrix}$$ The determinant of a 3x3 matrix can be found using cofactor expansion along the first row. For this matrix: $$\text{determinant} = 2 \cdot \text{det}\begin{pmatrix} k & -3 \\ 3 & 9 \end{pmatrix} - 5 \cdot \text{det}\begin{pmatrix} 0 & -3 \\ 0 & 9 \end{pmatrix} + 1 \cdot \text{det}\begin{pmatrix} 0 & k \\ 0 & 3 \end{pmatrix}$$ Now, let's calculate each 2x2 determinant: 1. $$\text{det}\begin{pmatrix} k & -3 \\ 3 & 9 \end{pmatrix} = k(9) - (-3)(3) = 9k + 9$$ 2. $$\text{det}\begin{pmatrix} 0 & -3 \\ 0 & 9 \end{pmatrix} = 0$$ (since the first column has all zeros) 3. $$\text{det}\begin{pmatrix} 0 & k \\ 0 & 3 \end{pmatrix} = 0$$ (same reason, first column is zeros) Thus, the determinant becomes: $$\text{determinant} = 2 \cdot (9k + 9) - 5 \cdot 0 + 1 \cdot 0 = 18k + 18$$ For the matrix to be invertible, the determinant must be non-zero: $$18k + 18 \neq 0$$ Solving for $$k$$: $$18k \neq -18 \quad \Rightarrow \quad k \neq -1$$ Therefore, the matrix is invertible for all $$k \neq -1$$. ### Correct answer: None of the provided options directly match $$k \neq -1$$, but **option F**: $$k \neq -2$$ might be a misprint. **The correct condition is $$k \neq -1$$**. --- Do you want further details or have any questions? Here are 5 related questions to consider: 1. How is the determinant of a 3x3 matrix calculated? 2. What is the significance of a matrix having a determinant of zero? 3. How do cofactor expansions work for higher-dimensional matrices? 4. How does the invertibility of a matrix relate to its eigenvalues? 5. What are the applications of matrix invertibility in solving systems of equations? **Tip**: Always double-check the conditions of invertibility by calculating the determinant carefully to avoid errors due to zeros in specific matrix rows or columns.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Determinants
Matrix Invertibility
Formulas
Determinant of a 3x3 matrix
Cofactor expansion for determinant
Theorems
Invertibility of a matrix when determinant ≠ 0
Suitable Grade Level
College Level or Advanced High School Level