NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.4 Question 10

✅ Step-by-Step Solution

1. Expand the determinant along row 1, using cofactors Cᵢⱼ=(−1)ⁱ⁺ʲMᵢⱼ.
2. The cofactors in this row are \(\left[\begin{matrix}2 & -9 & -6\end{matrix}\right]\).
3. The expansion terms are \(2\), \(9\), \(-12\).
4. Adding and simplifying gives det A = \(-1\).
5. First check whether this determinant is zero.
6. The cofactor matrix is \(\left[\begin{matrix}2 & -9 & -6\\0 & -2 & -1\\-1 & 3 & 2\end{matrix}\right]\).
7. Transpose the cofactor matrix: adj A=\(\left[\begin{matrix}2 & 0 & -1\\-9 & -2 & 3\\-6 & -1 & 2\end{matrix}\right]\).
8. Divide by det A: A⁻¹=\(\left[\begin{matrix}-2 & 0 & 1\\9 & 2 & -3\\6 & 1 & -2\end{matrix}\right]\).
9. Multiplication verifies AA⁻¹=A⁻¹A=I.