If A -122212221- then prove that A2-4 A-5 I-O-

Given: #8201; #160;A=122212221Now,A2=AA #8658;A2=122212221122212221 #8658;A2=1+4+42+2+42+4+22+2+44+1+44+2+22+4+24+2+24+4+1 #8658;A2=988898889A2 4A 5I ...

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Byju's Answer

Standard XII

Mathematics

Difference of Two Sets

If A =1222122...

Question

If $A = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}]$ , then prove that A² − 4A − 5I = O.

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Solution

$Given : A = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}] N o w, A^{2} = A A \Rightarrow A^{2} = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}] [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 1 + 4 + 4 & 2 + 2 + 4 & 2 + 4 + 2 \\ 2 + 2 + 4 & 4 + 1 + 4 & 4 + 2 + 2 \\ 2 + 4 + 2 & 4 + 2 + 2 & 4 + 4 + 1 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{matrix}] A^{2} - 4 A - 5 I \Rightarrow A^{2} - 4 A - 5 I = [\begin{matrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{matrix}] - 4 [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}] - 5 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \Rightarrow A^{2} - 4 A - 5 I = [\begin{matrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{matrix}] - [\begin{matrix} 4 & 8 & 8 \\ 8 & 4 & 8 \\ 8 & 8 & 4 \end{matrix}] - [\begin{matrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{matrix}] \Rightarrow A^{2} - 4 A - 5 I = [\begin{matrix} 9 - 4 - 5 & 8 - 8 - 0 & 8 - 8 - 0 \\ 8 - 8 - 0 & 9 - 4 - 5 & 8 - 8 - 0 \\ 8 - 8 - 0 & 8 - 8 - 0 & 9 - 4 - 5 \end{matrix}] \Rightarrow A^{2} - 4 A - 5 I = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] = 0$

Hence proved.

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Similar questions

Q. If

A = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}]

, find

A^{- 1}

and prove that

A^{2} - 4 A - 5 I = O

Q. If

A

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 2 2 & 1 & 2 2 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦

, then prove that

A^{2} - 4 A - 5 I = 0

Q. If

A = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 2 2 & 1 & 2 2 & 2 & 1 \end{matrix} ∣ ∣ ∣ ∣

, then prove that

A^{2} - 4 A - 5 I_{3} = 0

and hence obtain

A^{- 1}

Q. If matrix

A = [\begin{matrix} 1 & - 1 2 & 3 \end{matrix}]

, then prove that

A^{2} - 4 A + 5 I = 0

, where

I

is a unitary matrix.

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 2 2 & 1 & 2 2 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦

, then show that

A^{2} - 4 A - 5 I = 0

, and hence find

A^{- 1}

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If A=122212221, then prove that A2 − 4A − 5I = O.

If $A = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}]$ , then prove that A² − 4A − 5I = O.