Math Problem Statement
עבור אלו ערכי x המטריצה לא הפיכה
Solution
To determine for which values of the given matrix is non-invertible, we need to calculate the determinant of the matrix. A matrix is non-invertible if and only if its determinant is zero.
The given matrix:
The determinant of a matrix is given by:
Step 1: Compute the determinant
Substitute the elements of matrix : The determinant is:
Step 2: Expand the determinant
Expand the first term:
Expand the second term:
So the determinant becomes:
Simplify:
Step 3: Set the determinant to zero
To find the values of for which the matrix is non-invertible, solve:
Step 4: Factorize or solve the cubic equation
Factorize the cubic equation if possible. Let's try factoring by grouping or finding rational roots using the Rational Root Theorem.
The constant term is , and the leading coefficient is . Possible rational roots are factors of , which include:
Testing : Thus, is a root.
Step 5: Factorize the polynomial
Since is a root, divide the cubic polynomial by using synthetic or polynomial division.
Synthetic division:
-9 & 1 & 18 & -81 & -1458 \\ & & -9 & -81 & 729 \\ \hline & 1 & 9 & -162 & -729 \\ \end{array}$$ The quotient is: $$x^2 + 9x - 162$$ Thus, the polynomial factors as: $$x^3 + 18x^2 - 81x - 1458 = (x + 9)(x^2 + 9x - 162)$$ ### Step 6: Solve the quadratic factor Solve $$x^2 + 9x - 162 = 0$$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $$a = 1$$, $$b = 9$$, $$c = -162$$: $$x = \frac{-9 \pm \sqrt{9^2 - 4(1)(-162)}}{2(1)}$$ $$x = \frac{-9 \pm \sqrt{81 + 648}}{2}$$ $$x = \frac{-9 \pm \sqrt{729}}{2}$$ $$x = \frac{-9 \pm 27}{2}$$ Two solutions: $$x = \frac{-9 + 27}{2} = 9, \quad x = \frac{-9 - 27}{2} = -18$$ ### Final Answer: The matrix $$A$$ is non-invertible for: $$x = -9, \quad x = 9, \quad x = -18$$ Would you like a detailed explanation of the steps or further assistance? --- Here are 5 related questions: 1. How is the determinant of a $$2 \times 2$$ matrix computed? 2. Why does the determinant being zero imply that a matrix is non-invertible? 3. Can the Rational Root Theorem be applied to all polynomials? 4. How do you perform synthetic division step by step? 5. What is the significance of factoring cubic equations? **Tip:** Always check for simple roots of a polynomial first before attempting lengthy algebraic techniques!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Cubic and Quadratic Equations
Formulas
Determinant of a 2x2 matrix: det(A) = ad - bc
Cubic and quadratic equations: ax^3 + bx^2 + cx + d = 0, ax^2 + bx + c = 0
Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
Theorems
A matrix is non-invertible if and only if its determinant is zero.
Rational Root Theorem: Possible rational roots are factors of the constant term divided by factors of the leading coefficient.
Suitable Grade Level
Grades 11-12 or introductory university level
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