A-2-0-1-2-h-and-k-are-numbers-so-that-A-2-hA-kI-where-I-1-0-0-1-find-the-value-of-h-and-k-

Question Number 32869 by Rio Mike last updated on 04/Apr/18

A = (\begin{matrix} 2 0 \\ 1 2 \end{matrix}); h and k are numbers so

that A^{2} = h A + k I, where I = (\begin{matrix} 1 0 \\ 0 1 \end{matrix}) .

find the value of h and k .

Commented by abdo mathsup last updated on 09/Apr/18

t h e c s f a c t e t i s t i c p o l y n o m i a l o f A i s

p_{A} (X) = d e t (A - X I) = | \begin{matrix} 2 - X 0 \\ 1 2 - X \end{matrix} |

= {(2 - X)}^{2} = 4 - 4 X + X^{2} a n d k a y l e y h s m i l t o n

t h e o r e m g i v e p_{A} (A) = 0 \Rightarrow 4 I - 4 A + A^{2} = 0 \Rightarrow

A^{2} = 4 A - 4 I \Rightarrow h = 4 a n d k = - 4 .

Answered by Joel578 last updated on 05/Apr/18

A^{2} = (\begin{matrix} 2 & 0 \\ 1 & 2 \end{matrix}) (\begin{matrix} 2 & 0 \\ 1 & 2 \end{matrix}) = (\begin{matrix} 4 & 0 \\ 4 & 4 \end{matrix})

(\begin{matrix} 4 & 0 \\ 4 & 4 \end{matrix}) = (\begin{matrix} 2 h & 0 \\ h & 2 h \end{matrix}) + (\begin{matrix} k & 0 \\ 0 & k \end{matrix}) = (\begin{matrix} 2 h + k & 0 \\ h & 2 h + k \end{matrix})

h = 4

2 h + k = 4 \to k = - 4

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