A = 2 3 3

3 2 3

3 3 2 is matrix of order 3 * 3.

S.T : A^2 - 7A = 7I .Find A inverse ( without using any elementary transformation or adjoint method )

The question should be A² - 7A = 8I

A = [\begin{matrix} 2 & 3 & 3 \\ 3 & 2 & 3 \\ 3 & 3 & 2 \end{matrix}] A^{2} = [\begin{matrix} 2 & 3 & 3 \\ 3 & 2 & 3 \\ 3 & 3 & 2 \end{matrix}] \times [\begin{matrix} 2 & 3 & 3 \\ 3 & 2 & 3 \\ 3 & 3 & 2 \end{matrix}] = [\begin{matrix} 4 + 9 + 9 & 6 + 6 + 9 & 6 + 9 + 6 \\ 6 + 6 + 9 & 9 + 4 + 9 & 9 + 6 + 6 \\ 6 + 9 + 6 & 9 + 6 + 6 & 9 + 9 + 4 \end{matrix}] = [\begin{matrix} 22 & 21 & 21 \\ 21 & 22 & 21 \\ 21 & 21 & 22 \end{matrix}] L . H . S . = A^{2} - 7 A = [\begin{matrix} 22 & 21 & 21 \\ 21 & 22 & 21 \\ 21 & 21 & 22 \end{matrix}] - 7 [\begin{matrix} 2 & 3 & 3 \\ 3 & 2 & 3 \\ 3 & 3 & 2 \end{matrix}] = [\begin{matrix} 22 & 21 & 21 \\ 21 & 22 & 21 \\ 21 & 21 & 22 \end{matrix}] - [\begin{matrix} 14 & 21 & 21 \\ 21 & 14 & 21 \\ 21 & 21 & 14 \end{matrix}] = [\begin{matrix} 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{matrix}] = 8 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = 8 I = R . H . S . H e n c e P r o v e d . T o f i n d A^{- 1}, p r e m u l t i p l y b y A^{- 1}, w e h a v e, A^{- 1} (A^{2} - 7 A) = A^{- 1} (8 I) \Rightarrow A - 8 I = 8 A^{- 1} \Rightarrow A^{- 1} = \frac{1}{8} (A - 8 I) \Rightarrow A^{- 1} = \frac{1}{8} ([\begin{matrix} 2 & 3 & 3 \\ 3 & 2 & 3 \\ 3 & 3 & 2 \end{matrix}] - 8 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]) \Rightarrow A^{- 1} = \frac{1}{8} ([\begin{matrix} 2 & 3 & 3 \\ 3 & 2 & 3 \\ 3 & 3 & 2 \end{matrix}] - [\begin{matrix} 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{matrix}]) \Rightarrow A^{- 1} = \frac{1}{8} [\begin{matrix} - 6 & 3 & 3 \\ 3 & - 6 & 3 \\ 3 & 3 & - 6 \end{matrix}] = \frac{3}{8} [\begin{matrix} - 2 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix}] Hence, the answer .

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A = 2 3 3 3 2 3 3 3 2 is matrix of order 3 * 3. S.T : A^2 - 7A = 7I .Find A inverse ( without using any elementary transformation or adjoint method )

A = 2 3 3

3 2 3

3 3 2 is matrix of order 3 * 3.

S.T : A^2 - 7A = 7I .Find A inverse ( without using any elementary transformation or adjoint method )