Matrices
NCERT Class 12
CBSE
Grade 12
Question:
If $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$, find $A^2$ and show that $A^3 - 3A^2 - A + 9I = O$. Use this matrix equation to find $A^{-1}$.
Step-by-Step Solution
Calculate $A^2 = \begin{bmatrix} 0 & -1 & 5 \\ 1 & -4 & -4 \\ 4 & 1 & 1 \end{bmatrix}$ and $A^3 \Rightarrow A^3 - 3A^2 - A + 9I = O$. [2.0 Marks]
Multiply equation by $A^{-1}$: $9A^{-1} = -A^2 + 3A + I$. [1.5 Marks]
$9A^{-1} = \begin{bmatrix} 4 & -2 & -2 \\ 5 & 8 & -5 \\ -1 & 2 & 3 \end{bmatrix} \Rightarrow A^{-1} = \dfrac{1}{9} \begin{bmatrix} 4 & -2 & -2 \\ 5 & 8 & -5 \\ -1 & 2 & 3 \end{bmatrix}$. [1.5 Marks]
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🎯 Official CBSE Marking Scheme:
Evaluating $A^2, A^3$ & proving polynomial $O$: 2.0 Marks
Formulating inverse relation $9A^{-1} = -A^2 + 3A + I$: 1.5 Marks
Evaluating inverse matrix $A^{-1}$: 1.5 Marks
Correct Answer:
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