Determinants
NCERT Class 12
CBSE
Grade 12
Question:
If $A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$, verify that $A^3 - 6A^2 + 9A - 4I = O$ and hence find $A^{-1}$.
Step-by-Step Solution
Evaluate $A^2, A^3$ and prove $A^3 - 6A^2 + 9A - 4I = O$. [2.0 Marks]
Multiply by $A^{-1} \Rightarrow 4A^{-1} = A^2 - 6A + 9I$. [1.5 Marks]
$A^{-1} = \dfrac{1}{4} \begin{bmatrix} 3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix}$. [1.5 Marks]
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🎯 Official CBSE Marking Scheme:
Evaluating matrix powers $A^2, A^3$ and proving polynomial: 2.0 Marks
Formulating inverse relation $4A^{-1} = A^2 - 6A + 9I$: 1.5 Marks
Evaluating inverse matrix $A^{-1}$: 1.5 Marks
Correct Answer:
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