Math Problem Statement
Find the characteristic polynomial of A. Use x for the variable in your polynomial. You do not need to factor your polynomial.
Solution
To find the characteristic polynomial of the given matrix , we compute the determinant of , where is the identity matrix of the same size as and is a variable. Let's work through this step-by-step.
Matrix :
-5 & -2 & 0 \\ 12 & 5 & 0 \\ 12 & 4 & -3 \end{bmatrix}$$ ### $$A - xI$$: Subtract $$x$$ times the identity matrix from $$A$$: $$A - xI = \begin{bmatrix} -5 - x & -2 & 0 \\ 12 & 5 - x & 0 \\ 12 & 4 & -3 - x \end{bmatrix}$$ ### Characteristic Polynomial: The characteristic polynomial is given by: $$\text{det}(A - xI)$$ #### Determinant Calculation: Expanding along the first row: $$\text{det}(A - xI) = \begin{vmatrix} -5-x & -2 & 0 \\ 12 & 5-x & 0 \\ 12 & 4 & -3-x \end{vmatrix}$$ The determinant simplifies because the third column contains a $$0$$, meaning the expansion only involves the first two rows: $$\text{det}(A - xI) = (-5-x) \cdot \begin{vmatrix} 5-x & 0 \\ 4 & -3-x \end{vmatrix} - (-2) \cdot \begin{vmatrix} 12 & 0 \\ 12 & -3-x \end{vmatrix}$$ #### Sub-matrix Determinants: 1. For the first term: $$\begin{vmatrix} 5-x & 0 \\ 4 & -3-x \end{vmatrix} = (5-x)(-3-x) - (0)(4) = (-3-x)(5-x)$$ $$= -15 + 3x - 5x - x^2 = -x^2 - 2x - 15$$ 2. For the second term: $$\begin{vmatrix} 12 & 0 \\ 12 & -3-x \end{vmatrix} = (12)(-3-x) - (0)(12) = -36 - 12x$$ #### Substitute Back: $$\text{det}(A - xI) = (-5-x)(-x^2 - 2x - 15) - (-2)(-36 - 12x)$$ Expand both terms: 1. Expanding the first: $$(-5-x)(-x^2 - 2x - 15) = (-5)(-x^2) + (-5)(-2x) + (-5)(-15) + (-x)(-x^2) + (-x)(-2x) + (-x)(-15)$$ $$= 5x^2 + 10x + 75 + x^3 + 2x^2 + 15x$$ $$= x^3 + 7x^2 + 25x + 75$$ 2. Expanding the second: $$(-2)(-36 - 12x) = 72 + 24x$$ #### Combine: $$\text{det}(A - xI) = (x^3 + 7x^2 + 25x + 75) - (72 + 24x)$$ $$= x^3 + 7x^2 + (25x - 24x) + (75 - 72)$$ $$= x^3 + 7x^2 + x + 3$$ ### Final Answer: The characteristic polynomial is: $$x^3 + 7x^2 + x + 3$$ Would you like further details or assistance with understanding this calculation? Here are some questions that may interest you: 1. How do we interpret the roots of the characteristic polynomial? 2. Why is the characteristic polynomial important in linear algebra? 3. Can we solve this cubic polynomial for its eigenvalues? 4. What is the role of the identity matrix in the calculation? 5. How does the determinant expand based on the cofactor method? **Tip:** Always check if simplifying assumptions, such as zeros in rows or columns, can make determinant calculationsAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Eigenvalues
Characteristic Polynomial
Formulas
Characteristic Polynomial: det(A - xI)
Theorems
Cayley-Hamilton Theorem
Determinant Expansion by Cofactors
Suitable Grade Level
College Level (Linear Algebra Course)