CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Q364 Marksmatrix_linear_systems
CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Question 36 (Matrices)
A cuboid’s length plus breadth exceeds its height by 3 cm. Twice its length plus three times its breadth plus its height is 10 cm. Its breadth plus seven times its height is one less than three times its length.
Part (1) 1 Marks
Write the equations in the form AX=B.
Answer: [[1,1,−1],[2,3,1],[−3,1,7]](l,b,h)ᵀ=(3,10,−1)ᵀ.
- Rearrange to l+b−h=3, 2l+3b+h=10, and −3l+b+7h=−1.
- Use their coefficients as the rows of A in the order l,b,h.
Part (2) 1 Marks
Does A⁻¹ exist? Justify.
Answer: Yes, det A=−8.
- Expand along the first row: det A=20−17−11=−8.
- Since it is nonzero, A is invertible.
Part (3) 2 Marks
Find A⁻¹.
Answer: [[20,−8,4],[−17,4,−3],[11,−4,1]]/(−8).
- Compute the cofactor matrix of A.
- Transpose it to obtain adj(A)=[[20,−8,4],[−17,4,−3],[11,−4,1]].
- Divide by det A=−8 to obtain A⁻¹.
Part (4) 2 Marks
Find A²+7I.
Answer: [[13,3,−7],[5,19,8],[−22,7,60]].
- Multiply the coefficient matrix A by itself.
- The product is [[6,3,−7],[5,12,8],[−22,7,53]].
- Increase each diagonal entry by seven.