Math Problem Statement

עבור אלו ערכי x המטריצה לא הפיכה

עבור אלו ערכי x המטריצה לא הפיכה

Solution

To determine for which values of xx the given matrix AA 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:

A=(x28171622x2x+4)A = \begin{pmatrix} x^2 - 81 & 7 \\ 162 - 2x^2 & x + 4 \end{pmatrix}

The determinant of a 2×22 \times 2 matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} is given by: det(A)=adbc\text{det}(A) = ad - bc

Step 1: Compute the determinant

Substitute the elements of matrix AA: a=x281,b=7,c=1622x2,d=x+4a = x^2 - 81, \, b = 7, \, c = 162 - 2x^2, \, d = x + 4 The determinant is: det(A)=(x281)(x+4)(7)(1622x2)\text{det}(A) = (x^2 - 81)(x + 4) - (7)(162 - 2x^2)

Step 2: Expand the determinant

Expand the first term: (x281)(x+4)=x3+4x281x324(x^2 - 81)(x + 4) = x^3 + 4x^2 - 81x - 324

Expand the second term: 7(1622x2)=113414x27(162 - 2x^2) = 1134 - 14x^2

So the determinant becomes: det(A)=x3+4x281x324(113414x2)\text{det}(A) = x^3 + 4x^2 - 81x - 324 - (1134 - 14x^2)

Simplify: det(A)=x3+4x281x3241134+14x2\text{det}(A) = x^3 + 4x^2 - 81x - 324 - 1134 + 14x^2 det(A)=x3+18x281x1458\text{det}(A) = x^3 + 18x^2 - 81x - 1458

Step 3: Set the determinant to zero

To find the values of xx for which the matrix is non-invertible, solve: x3+18x281x1458=0x^3 + 18x^2 - 81x - 1458 = 0

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 1458-1458, and the leading coefficient is 11. Possible rational roots are factors of 1458-1458, which include: ±1,±2,±3,±6,±9,±27,±54,±81,±162,±486,±729,±1458\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 27, \pm 54, \pm 81, \pm 162, \pm 486, \pm 729, \pm 1458

Testing x=9x = -9: (9)3+18(9)281(9)1458=729+1458+7291458=0(-9)^3 + 18(-9)^2 - 81(-9) - 1458 = -729 + 1458 + 729 - 1458 = 0 Thus, x=9x = -9 is a root.

Step 5: Factorize the polynomial

Since x=9x = -9 is a root, divide the cubic polynomial x3+18x281x1458x^3 + 18x^2 - 81x - 1458 by x+9x + 9 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

By Image

Drop file here or Click Here to upload

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