Math Problem Statement
The value(s) of k for which the matrix -1 1 k 0 1 4 K 3 7 Is nonsingular is / are given by:
- K is not equal to -1 or k is not equal to 0
- K is not equal to -1
- K is not equal 5 or k is not equal to 1
- K is not equal to 5 and k is not equal to 1
- K is not equal 5
- None of the given option
- K is not equal -1 or k is not equal to 2
Solution
To determine the values of and for which the given matrix is nonsingular, we need to ensure that its determinant is not equal to zero.
The given matrix is:
-1 & 1 & k \\ 0 & 1 & 4 \\ K & 3 & 7 \end{bmatrix}$$ ### Step 1: Compute the determinant The determinant of a $$3 \times 3$$ matrix is given by: $$\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)$$ For this matrix: $$\text{det} = (-1) \cdot (1 \cdot 7 - 4 \cdot 3) - 1 \cdot (0 \cdot 7 - 4 \cdot K) + k \cdot (0 \cdot 3 - 1 \cdot K)$$ Simplify each term: - First term: $$(-1)(7 - 12) = (-1)(-5) = 5$$ - Second term: $$- (0 - 4K) = -(-4K) = 4K$$ - Third term: $$k (0 - K) = -kK$$ Thus: $$\text{det} = 5 + 4K - kK$$ ### Step 2: Matrix is nonsingular A matrix is nonsingular if its determinant is not zero. Thus, for nonsingularity: $$5 + 4K - kK \neq 0$$ This inequality implies that specific combinations of $$K$$ and $$k$$ will make the determinant zero, and these values must be avoided. ### Step 3: Analyze options The question provides possible conditions for $$K$$ and $$k$$. Substituting and simplifying for each, we identify the conditions for nonsingularity: 1. $$K \neq -1$$ or $$k \neq 0$$: Does not guarantee nonsingularity for all $$K, k$$. 2. $$K \neq -1$$: This is partially relevant but incomplete. 3. $$K \neq 5$$ or $$k \neq 1$$: Appears promising since $$kK$$ creates singularity at certain values. 4. $$K \neq 5$$ and $$k \neq 1$$: Correct, avoids determinant equal to zero. 5. $$K \neq 5$$: Incomplete condition. 6. None of the given options: Likely incorrect since one option matches. 7. $$K \neq -1$$ or $$k \neq 2$$: Incorrect for specific configurations. ### Final Answer: **Option 4: $$K \neq 5$$ and $$k \neq 1$$.** Would you like a detailed explanation or numerical verification? --- 1. What does nonsingular mean in terms of matrices? 2. How does changing $$k$$ affect matrix singularity? 3. Why is determinant computation critical for linear algebra problems? 4. Can $$K$$ and $$k$$ take fractional values here? 5. What other scenarios might make a matrix singular? **Tip:** Always check edge cases when working with determinants to ensure no overlooked configurations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Theory
Determinants
Linear Algebra
Formulas
Determinant of a 3x3 matrix: det = a(ei - fh) - b(di - fg) + c(dh - eg)
Theorems
A matrix is nonsingular if its determinant is not zero.
Determinant properties for matrix invertibility
Suitable Grade Level
Undergraduate or advanced high school (Grades 11-12)