NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.4 Question 8

✅ Step-by-Step Solution

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