Class 12 Matrices · Short Answer2 Markssymmetric_skew_decomposition
Class 12 Maths · Matrices · Short Answer #91
Express A=\(\begin{bmatrix}4 & -3\\2 & -1\end{bmatrix}\) as a sum of symmetric and skew-symmetric matrices.
✅ Step-by-Step Solution
Final Answer: P=\(\begin{bmatrix}4 & -1/2\\-1/2 & -1\end{bmatrix}\); Q=\(\begin{bmatrix}0 & -5/2\\5/2 & 0\end{bmatrix}\)
- Use P=(A+Aᵀ)/2 and Q=(A−Aᵀ)/2.
- The two parts are P=\(\begin{bmatrix}4 & -1/2\\-1/2 & -1\end{bmatrix}\) and Q=\(\begin{bmatrix}0 & -5/2\\5/2 & 0\end{bmatrix}\). Their transposes satisfy Pᵀ=P and Qᵀ=−Q.
- Thus A=P+Q with the displayed parts, whose entries sum back to A.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Use P=(A+Aᵀ)/2 and Q=(A−Aᵀ)/2.
- [1 mark] The two parts are P=\(\begin{bmatrix}4 & -1/2\\-1/2 & -1\end{bmatrix}\) and Q=\(\begin{bmatrix}0 & -5/2\\5/2 & 0\end{bmatrix}\). Their transposes satisfy Pᵀ=P and Qᵀ=−Q.
- [0.5 marks] Thus A=P+Q with the displayed parts, whose entries sum back to A.
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