Mathbee CBSE Mathbee · CBSE Class 12 Maths
Class 12 Matrices · Long Answer3 Markssymmetric_skew_decomposition

Class 12 Maths · Matrices · Long Answer #100

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

✅ Step-by-Step Solution

Final Answer: P=\(\begin{bmatrix}2 & 2 & 5/2\\2 & -1 & 3/2\\5/2 & 3/2 & 2\end{bmatrix}\); Q=\(\begin{bmatrix}0 & 1 & -3/2\\-1 & 0 & 1/2\\3/2 & -1/2 & 0\end{bmatrix}\)
  1. Take the canonical parts P=(A+Aᵀ)/2 and Q=(A−Aᵀ)/2.
  2. Calculation gives P=\(\begin{bmatrix}2 & 2 & 5/2\\2 & -1 & 3/2\\5/2 & 3/2 & 2\end{bmatrix}\) and Q=\(\begin{bmatrix}0 & 1 & -3/2\\-1 & 0 & 1/2\\3/2 & -1/2 & 0\end{bmatrix}\).
  3. Transposition gives Pᵀ=P and Qᵀ=−Q; adding the two displayed matrices gives A entrywise.
  4. Thus the required decomposition is A=P+Q.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Take the canonical parts P=(A+Aᵀ)/2 and Q=(A−Aᵀ)/2.
  • [1 mark] Calculation gives P=\(\begin{bmatrix}2 & 2 & 5/2\\2 & -1 & 3/2\\5/2 & 3/2 & 2\end{bmatrix}\) and Q=\(\begin{bmatrix}0 & 1 & -3/2\\-1 & 0 & 1/2\\3/2 & -1/2 & 0\end{bmatrix}\).
  • [1 mark] Transposition gives Pᵀ=P and Qᵀ=−Q; adding the two displayed matrices gives A entrywise.
  • [0.5 marks] Thus the required decomposition is A=P+Q.

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}1 & 0 & 2\\0 & 2 & 1\\2 & 0 &…
Next Question →
For A=\(\begin{bmatrix}2 & 0 & 1\\2 & 1 & 3\\1 & -1 &…

More Practice Questions in Matrices

Q1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q12Q13Q14Q15Q16Q17Q18Q19Q20Q21Q22Q23Q24Q25Q26Q27Q28Q29Q30Q31Q32Q33Q34Q35Q36Q37Q38Q39Q40

Related Questions in Matrices

Class 12 Matrices · Mcq Single Correct
Let A be a 2×2 matrix with aᵢⱼ=1 for i≠j and 0 for i=j. Find A².
Class 12 Matrices · Mcq Single Correct
Let A=\(\begin{bmatrix}\cos\alpha & -\sin\alpha\\\sin\alpha & \cos\alpha\end{bmatrix}\)…
Class 12 Matrices · Mcq Single Correct
If \(\begin{bmatrix}2a+b & a−2b\\5c−d & 4c+3d\end{bmatrix}\)=\(\begin{bmatrix}4 & -3\\11 &…
Class 12 Matrices · Mcq Single Correct
How many 3×3 matrices have every entry either 2 or 0?
Class 12 Matrices · Mcq Single Correct
Find x+y for constants x,y such that the identity A⁻¹=xA+yI holds for every real square…
Class 12 Matrices · Mcq Single Correct
If \(\begin{bmatrix}2 & 0\\5 & 4\end{bmatrix}\)=P+Q, where P is symmetric and Q…