Suppose A is a 3 × 3 matrix such that det(A − λI) = −λ^3 + 3λ^2 + λ − 3, where I is the 3 × 3 identity matrix. Which of the following are TRUE of A? I. The trace of A is 3. II. The determinate of A is −3. III. The matrix A has eigenvalues −3 and 1.

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the characteristic polynomial of matrix A. ‖ ‖ step 2 ⋮ The characteristic polynomial is given by $ \text{det}(A - \lambda I) = -\lambda^3 + 3\lambda^2 + \lambda - 3 $. ‖ ‖ step 3 ⋮ The trace of a matrix is the sum of its eigenvalues, which is the sum of the roots of its characteristic polynomial. ‖ ‖ step 4 ⋮ The sum of the roots of the characteristic polynomial is given by the coefficient of the $\lambda^2$ term with the opposite sign, which is 3. ‖ ‖ step 5 ⋮ The determinant of a matrix is the product of its eigenvalues, which is the constant term of its characteristic polynomial. ‖ ‖ step 6 ⋮ The constant term of the characteristic polynomial is -3, which is the determinant of matrix A. ‖ ‖ step 7 ⋮ To find the eigenvalues, we need to solve the characteristic polynomial equation $ -\lambda^3 + 3\lambda^2 + \lambda - 3 = 0 $. ‖ ‖ step 8 ⋮ By inspection, $\lambda = 1$ is a root of the polynomial because $ -(1)^3 + 3(1)^2 + (1) - 3 = 0 $. ‖ ‖ step 9 ⋮ To find the other roots, we can perform polynomial division or use synthetic division to divide the characteristic polynomial by $(\lambda - 1)$. ‖ ‖ step 10 ⋮ After dividing, we get a quadratic polynomial. We can then use the quadratic formula to find the remaining roots. ‖ ‖ step 11 ⋮ If $\lambda = -3$ is another root, it should satisfy the characteristic polynomial equation. ‖ ‖ step 12 ⋮ Substituting $\lambda = -3$ into the polynomial, we get $ -(-3)^3 + 3(-3)^2 + (-3) - 3 $, which simplifies to $ -27 + 27 - 3 - 3 $. ‖ ‖ step 13 ⋮ Simplifying the expression, we get $ -27 + 27 - 3 - 3 = -6 $, which is not equal to 0, so $\lambda = -3$ is not a root. ‖ ∻Answer∻ ⚹ Statements I and II are TRUE, while statement III is FALSE. ⚹ ∻Key Concept∻ ⚹Characteristic Polynomial and Matrix Properties⚹ ∻Explanation∻ ⚹The trace of a matrix is the sum of its eigenvalues, which corresponds to the coefficient of the second highest degree term in the characteristic polynomial with the opposite sign. The determinant of a matrix is the product of its eigenvalues, which is the constant term of the characteristic polynomial. To verify the eigenvalues, they must satisfy the characteristic polynomial equation.⚹◊What is the relationship between the eigenvalues of a matrix A and its characteristic polynomial det(A - λI)?⍭ Generate me a similar question◊

Sia