Math Problem Statement
Considere a seguinte matriz A = [[1, 1, k], [1, 2, 7], [1, 3, 1]]. Determine todos os valores de k que tornam a matriz A invertível.
Solution
The given problem asks to find the values of that make the matrix invertible. The matrix is:
1 & 1 & k \\ 1 & 2 & 7 \\ 1 & 3 & 1 \end{pmatrix}$$ To determine when a matrix is invertible, we need to check when its determinant is non-zero. The determinant of a $$3 \times 3$$ matrix is given by: $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ Where the matrix is: $$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ For matrix $$A$$, we have: $$\begin{pmatrix} 1 & 1 & k \\ 1 & 2 & 7 \\ 1 & 3 & 1 \end{pmatrix}$$ Thus: - $$a = 1$$ - $$b = 1$$ - $$c = k$$ - $$d = 1$$ - $$e = 2$$ - $$f = 7$$ - $$g = 1$$ - $$h = 3$$ - $$i = 1$$ Now, let's calculate the determinant. I'll do this calculation.The determinant of matrix $$A$$ is $$\text{det}(A) = k - 13$$. For the matrix $$A$$ to be invertible, its determinant must be non-zero. Therefore, $$k - 13 \neq 0$$, which gives: $$k \neq 13$$ Thus, the matrix $$A$$ is invertible for all values of $$k$$ except $$k = 13$$. Would you like more details or have any questions about this? Here are 5 related questions you could explore: 1. What happens to the matrix when $$k = 13$$? 2. How do you calculate the inverse of a $$3 \times 3$$ matrix? 3. How does the determinant help in determining the invertibility of a matrix? 4. What are some real-world applications of matrix invertibility? 5. How does row reduction relate to determining if a matrix is invertible? **Tip:** A matrix with a determinant of zero is called "singular" and cannot be inverted.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Determinants
Formulas
Determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
Theorems
Invertibility Theorem: A matrix is invertible if and only if its determinant is non-zero.
Suitable Grade Level
Undergraduate