CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q325 Marksmatrix_linear_systems
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Question 32 (Matrices)
For P=[[1,−1,0],[2,3,4],[0,1,2]] and Q=[[2,2,−4],[−4,2,−4],[2,−1,5]], compute QP. Use matrices to solve x−y=3, 2x+3y+4z=17, y+2z=7. OR Evaluate the determinant of [[1+x,1,1],[1,1+y,1],[1,1,1+z]]. If it vanishes and x,y,z are nonzero, prove 1/x+1/y+1/z=−1.
Alternative 1 5 Marks
For P=[[1,−1,0],[2,3,4],[0,1,2]] and Q=[[2,2,−4],[−4,2,−4],[2,−1,5]], compute QP. Use matrices to solve x−y=3, 2x+3y+4z=17, y+2z=7.
Answer: QP=6I and (x,y,z)=(2,−1,4).
- Multiply row by column to get QP=diag(6,6,6), so P⁻¹=Q/6.
- Writing P(x,y,z)ᵀ=(3,17,7)ᵀ, multiply by Q to get 6(x,y,z)ᵀ=(12,−6,24)ᵀ.
- Divide by 6 to obtain (2,−1,4), which satisfies all three equations.
Alternative 2 5 Marks
Evaluate the determinant of [[1+x,1,1],[1,1+y,1],[1,1,1+z]]. If it vanishes and x,y,z are nonzero, prove 1/x+1/y+1/z=−1.
Answer: Δ=xyz+xy+yz+zx. If Δ=0, 1/x+1/y+1/z=−1.
- Subtract row 1 from rows 2 and 3, giving rows (−x,y,0) and (−x,0,z).
- Expansion along row 1 gives (1+x)yz+xz+xy=xyz+yz+xz+xy.
- When Δ=0, divide by xyz to obtain the required reciprocal identity.