Matrix to Polynomial

Algebra Level 5

Let ρ : G L ( 2 , R ) → R [ x ] \rho: GL(2,\mathbb{R}) \rightarrow \mathbb{R}_{[x]} , ∀ A ∈ G L ( 2 , R ) \forall A\in GL(2,\mathbb{R}) we defined ρ ( A ) = d e t ( x I 2 − A ) \rho(A) = det(xI_{2} - A) . Find d e t ( A ) det(A) if d e t ( x I 2 − A ) = x 2 − 900 x + 100 det(xI_{2} - A) = x^2 -900x + 100 .

Details : A I 2 = A = I 2 A AI_{2} = A = I_{2}A


The answer is 100.

Paul Ryan Longhas by Paul Ryan Longhas , 24, Philippines

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2 solutions

Otto Bretscher
Dec 19, 2015

Let x = 0 x=0 in the equation det ⁡ ( x I 2 − A ) = x 2 − 900 x + 100 \det(xI_2-A)=x^2-900x+100 , noting that det ⁡ ( − A ) = det ⁡ ( A ) \det(-A)=\det(A) for a 2 × 2 2\times 2 matrix A A .

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Jun Arro Estrella
Dec 21, 2015

Characteristic polynomials..

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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