Math Problem Statement
Using only a simple calculator, find the values of
k
such that
det
(
M
)
0
, where
M
⎡
⎢
⎣
−
5
k
0
1
1
k
1
1
−
8
⎤
⎥
⎦
. As your answer, enter the SUM of the value(s) of
k
that satisfy this condition.
Solution
To solve for such that , we need to calculate the determinant of the matrix and set it equal to zero. The matrix is given as:
-5 & k & 0 \\ 1 & 1 & k \\ 1 & 1 & -8 \end{bmatrix}.$$ ### Step 1: Formula for the determinant of a 3x3 matrix The determinant of $$M$$ is computed as: $$\det(M) = a(ei − fh) − b(di − fg) + c(dh − eg),$$ where the matrix is: $$M = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}.$$ ### Step 2: Substitute the elements of $$M$$ Here: - $$a = -5, b = k, c = 0$$ - $$d = 1, e = 1, f = k$$ - $$g = 1, h = 1, i = -8$$. Substituting into the determinant formula: $$\det(M) = -5 \cdot (1 \cdot -8 - k \cdot 1) - k \cdot (1 \cdot -8 - k \cdot 1) + 0.$$ ### Step 3: Simplify each term First term: $$-5 \cdot (-8 - k) = -5 \cdot -8 - (-5) \cdot k = 40 + 5k.$$ Second term: $$-k \cdot (-8 - k) = -k \cdot -8 - k \cdot k = 8k - k^2.$$ Third term is zero: $$0.$$ So: $$\det(M) = 40 + 5k + 8k - k^2.$$ Combine like terms: $$\det(M) = -k^2 + 13k + 40.$$ ### Step 4: Set $$\det(M) = 0$$ Solve the quadratic equation: $$-k^2 + 13k + 40 = 0.$$ Multiply through by -1 to simplify: $$k^2 - 13k - 40 = 0.$$ ### Step 5: Factorize the quadratic Find two numbers that multiply to $$-40$$ and add to $$-13$$. These are $$-8$$ and $$5$$. So: $$k^2 - 13k - 40 = (k - 8)(k + 5) = 0.$$ ### Step 6: Solve for $$k$$ $$k - 8 = 0 \quad \text{or} \quad k + 5 = 0.$$ $$k = 8 \quad \text{or} \quad k = -5.$$ ### Step 7: Find the sum of $$k$$ The sum of $$k$$ values is: $$8 + (-5) = 3.$$ ### Final Answer: The sum of the values of $$k$$ is: $$\boxed{3}.$$ Would you like further clarification or details on any step? --- **Related Questions:** 1. How do you verify the determinant calculation for a specific value of $$k$$? 2. What happens if the determinant is nonzero for a matrix? 3. How do you graphically represent the solutions for $$k$$ in this case? 4. Can this matrix be diagonalized for certain values of $$k$$? 5. What is the geometric interpretation of $$\det(M) = 0$$? **Tip:** Always double-check factorization and substitution stepsAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Determinants
Quadratic Equations
Matrix Algebra
Formulas
Determinant of a 3x3 matrix: det(M) = a(ei − fh) − b(di − fg) + c(dh − eg)
Quadratic equation: ax^2 + bx + c = 0
Theorems
Fundamental Theorem of Algebra
Properties of Determinants
Suitable Grade Level
Grades 9-12