Math Problem Statement
Use the Invertible Matrix Theorem to decide if A is invertible:
A = [ [1, 0, -2], [3, 1, -2], [-5, -1, 9] ]
Solution
To determine if matrix is invertible, we use the Invertible Matrix Theorem, which states that a square matrix is invertible if and only if its determinant is non-zero. Let us calculate the determinant of the given matrix :
\begin{bmatrix} 1 & 0 & -2 \\ 3 & 1 & -2 \\ -5 & -1 & 9 \end{bmatrix}$$ ### Determinant Calculation The determinant of a $$3 \times 3$$ matrix is computed using the formula: $$\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)$$ where: $$A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}.$$ For this matrix: $$a = 1, \, b = 0, \, c = -2, \, d = 3, \, e = 1, \, f = -2, \, g = -5, \, h = -1, \, i = 9$$ Plugging into the determinant formula: $$\text{det}(A) = 1\big( (1)(9) - (-2)(-1) \big) - 0\big( (3)(9) - (-2)(-5) \big) + (-2)\big( (3)(-1) - (1)(-5) \big)$$ ### Step-by-step evaluation: 1. Compute each minor: - $$(1)(9) - (-2)(-1) = 9 - 2 = 7$$, - $$(3)(-1) - (1)(-5) = -3 + 5 = 2$$. 2. Substitute into the determinant formula: $$\text{det}(A) = 1(7) - 0 + (-2)(2)$$ $$\text{det}(A) = 7 - 4 = 3$$ ### Conclusion: The determinant of $$A$$ is $$\text{det}(A) = 3$$, which is non-zero. Thus, by the **Invertible Matrix Theorem**, $$A$$ is invertible. --- ### Would you like more details or have any questions? --- ### 5 Related Questions: 1. What are the implications of a determinant being zero? 2. How does the Invertible Matrix Theorem apply to $$n \times n$$ matrices generally? 3. Can we determine invertibility using row reduction instead of the determinant? 4. What does the determinant represent geometrically for a $$3 \times 3$$ matrix? 5. How can we compute the inverse of matrix $$A$$ if it is invertible? --- ### Tip: To quickly check for invertibility without calculating the full determinant, look for properties such as linear dependence of rows or columns, which implies a zero determinant.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrix Invertibility
Formulas
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Invertible Matrix Theorem
Suitable Grade Level
Undergraduate (Introductory Linear Algebra)