Class 12 Matrices · Long Answer3 Markssymmetric_skew_decomposition

Class 12 Maths · Matrices · Long Answer #69

Express A=\(\begin{bmatrix}3 & -2 & -4\\3 & -2 & -5\\-1 & 1 & 2\end{bmatrix}\) as the sum of a symmetric matrix and a skew-symmetric matrix, and verify.

✅ Step-by-Step Solution

Final Answer: P=\(\begin{bmatrix}3 & 1/2 & -5/2\\1/2 & -2 & -2\\-5/2 & -2 & 2\end{bmatrix}\); Q=\(\begin{bmatrix}0 & -5/2 & -3/2\\5/2 & 0 & -3\\3/2 & 3 & 0\end{bmatrix}\)

Use P=(A+Aᵀ)/2 and Q=(A−Aᵀ)/2.
Adding and subtracting the transpose gives P=\(\begin{bmatrix}3 & 1/2 & -5/2\\1/2 & -2 & -2\\-5/2 & -2 & 2\end{bmatrix}\) and Q=\(\begin{bmatrix}0 & -5/2 & -3/2\\5/2 & 0 & -3\\3/2 & 3 & 0\end{bmatrix}\).
Entrywise, Pᵀ=P and Qᵀ=−Q. Also P+Q=½[(A+Aᵀ)+(A−Aᵀ)]=A; the displayed entries sum to the given matrix.
Thus A=P+Q with the required symmetric and skew-symmetric parts.

📋 CBSE Stepwise Marking Scheme

[0.5 marks] Use P=(A+Aᵀ)/2 and Q=(A−Aᵀ)/2.
[1 mark] Adding and subtracting the transpose gives P=\(\begin{bmatrix}3 & 1/2 & -5/2\\1/2 & -2 & -2\\-5/2 & -2 & 2\end{bmatrix}\) and Q=\(\begin{bmatrix}0 & -5/2 & -3/2\\5/2 & 0 & -3\\3/2 & 3 & 0\end{bmatrix}\).
[1 mark] Entrywise, Pᵀ=P and Qᵀ=−Q. Also P+Q=½[(A+Aᵀ)+(A−Aᵀ)]=A; the displayed entries sum to the given matrix.
[0.5 marks] Thus A=P+Q with the required symmetric and skew-symmetric parts.

Master Chapter 3: Matrices

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Matrices.

📘 Chapter Notes & All Exercises 🚀 Open Interactive Workspace →

← Previous Question

For A=\(\begin{bmatrix}2 & 3\\-1 & 2\end{bmatrix}\), prove…

Next Question →

For A=\(\begin{bmatrix}1 & 2 & 3\\3 & -2 & 1\\4 & 2 &…

Class 12 Maths · Matrices · Long Answer #69

✅ Step-by-Step Solution

📋 CBSE Stepwise Marking Scheme

Master Chapter 3: Matrices

More Practice Questions in Matrices

Related Questions in Matrices