CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Q313 Marksmatrix_decomposition
CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Question 31 (Matrices)
Express \(A=\begin{bmatrix}2&4\\5&6\end{bmatrix}\) as a sum of a symmetric and a skew-symmetric matrix. OR Using matrices, find the intersection of 5x−7y=2 and 7x−5y=3.
Alternative 1 3 Marks
Express \(A=\begin{bmatrix}2&4\\5&6\end{bmatrix}\) as a sum of a symmetric and a skew-symmetric matrix.
Answer: P=[[2,9/2],[9/2,6]]; Q=[[0,−1/2],[1/2,0]].
- Use P=(A+Aᵀ)/2 and Q=(A−Aᵀ)/2.
- The off-diagonal average is 9/2 and the upper signed half-difference is −1/2. The diagonals of P are 2,6 and those of Q are zero.
- Thus the displayed P,Q have the required transpose properties and P+Q=A.
Alternative 2 3 Marks
Using matrices, find the intersection of 5x−7y=2 and 7x−5y=3.
Answer: (11/24,1/24)
- The coefficient matrix has determinant −25+49=24.
- Multiply its inverse [[−5,7],[−7,5]]/24 by (2,3)ᵀ to obtain (11,1)ᵀ/24.
- Hence the intersection is (11/24,1/24).