Write down the characteristic polynomial of A = [4 −2 1; 0 3 0; 2 −2 5] and report its largest root.

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Linear algebraHard

Write down the characteristic polynomial of $A = 402 - 2 3 - 2 105$ and report its largest root.

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Hint

Expand $det (A - λ I)$ along the row with the most zeros.

Solution

Row $2$ of $A - λ I$ is $(0, 3 - λ, 0)$ , so expanding along it gives $det (A - λ I) = (3 - λ) \times det ([4 - λ 2 1 5 - λ])$ .
The remaining determinant is $(4 - λ) (5 - λ) - 2 = λ^{2} - 9 λ + 18 = (λ - 3) (λ - 6)$ .
So the eigenvalues are $3$ , $3$ and $6$ , and their sum $3 + 3 + 6 = 12$ matches the trace $4 + 3 + 5 = 12$ .
The largest eigenvalue is $6$ .

Answer: $6$