Class 12 Matrices · Long Answer3 Marksmatrix_polynomials
Class 12 Maths · Matrices · Long Answer #99
For A=\(\begin{bmatrix}1 & 0 & 2\\0 & 2 & 1\\2 & 0 & 3\end{bmatrix}\), prove A³−6A²+7A+2I=O.
✅ Step-by-Step Solution
Final Answer: A³−6A²+7A+2I=O
- Compute A²=\(\begin{bmatrix}5 & 0 & 8\\2 & 4 & 5\\8 & 0 & 13\end{bmatrix}\).
- Multiply again to obtain A³=\(\begin{bmatrix}21 & 0 & 34\\12 & 8 & 23\\34 & 0 & 55\end{bmatrix}\).
- Combine the four terms entrywise: A³−6A²+7A+2I=\(\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\).
- The result is the zero 3×3 matrix, proving the identity.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Compute A²=\(\begin{bmatrix}5 & 0 & 8\\2 & 4 & 5\\8 & 0 & 13\end{bmatrix}\).
- [1 mark] Multiply again to obtain A³=\(\begin{bmatrix}21 & 0 & 34\\12 & 8 & 23\\34 & 0 & 55\end{bmatrix}\).
- [1 mark] Combine the four terms entrywise: A³−6A²+7A+2I=\(\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\).
- [0.5 marks] The result is the zero 3×3 matrix, proving the identity.
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