Math Problem Statement
Find all values of k so that the matrix \[ \begin{bmatrix} k+1 & 4 \\ 9 & k+1 \end{bmatrix} - I_2 \] is invertible.
Solution
Let's solve the problem.
The matrix given is:
k+1 & 4 \\ 9 & k+1 \end{bmatrix} - I_2,$$ where $$I_2$$ is the identity matrix: $$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$$ ### Subtracting $$I_2$$: $$\begin{bmatrix} k+1 & 4 \\ 9 & k+1 \end{bmatrix} - I_2 = \begin{bmatrix} k & 4 \\ 9 & k \end{bmatrix}.$$ The question asks for the values of $$k$$ for which this matrix is **invertible**. A matrix is invertible if its determinant is **non-zero**. ### Determinant of the matrix: $$\text{det} = \begin{vmatrix} k & 4 \\ 9 & k \end{vmatrix} = (k)(k) - (9)(4) = k^2 - 36.$$ ### Condition for invertibility: The determinant must not be zero: $$k^2 - 36 \neq 0.$$ ### Solving: $$k^2 \neq 36 \quad \Rightarrow \quad k \neq \pm 6.$$ ### Conclusion: The matrix is invertible for all values of $$k$$, **except** $$k = 6$$ and $$k = -6$$. ### Final Answer: - **Invertible**: When $$k \neq 6, -6$$. --- Would you like more details or further clarifications? 😊 Here are 5 related questions to expand your understanding: 1. What is the meaning of matrix invertibility in linear algebra? 2. How does subtracting the identity matrix affect the determinant? 3. Can a zero determinant ever indicate an invertible matrix? 4. What are the implications of non-invertibility in systems of linear equations? 5. How can eigenvalues help identify non-invertible matrices? **Tip**: To check matrix invertibility quickly, always calculate the determinant first!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Invertibility
Determinants
Formulas
det(A) = ad - bc
Condition for invertibility: det(A) ≠ 0
Theorems
Invertibility of a square matrix based on determinant
Suitable Grade Level
Grades 10-12