NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.4 Question 7

✅ Step-by-Step Solution

1. Expand the determinant along row 1, using cofactors Cᵢⱼ=(−1)ⁱ⁺ʲMᵢⱼ.
2. The cofactors in this row are \(\left[\begin{matrix}10 & 0 & 0\end{matrix}\right]\).
3. The expansion terms are \(10\), \(0\), \(0\).
4. Adding and simplifying gives det A = \(10\).
5. First check whether this determinant is zero.
6. The cofactor matrix is \(\left[\begin{matrix}10 & 0 & 0\\-10 & 5 & 0\\2 & -4 & 2\end{matrix}\right]\).
7. Transpose the cofactor matrix: adj A=\(\left[\begin{matrix}10 & -10 & 2\\0 & 5 & -4\\0 & 0 & 2\end{matrix}\right]\).
8. Divide by det A: A⁻¹=\(\left[\begin{matrix}1 & -1 & \frac{1}{5}\\0 & \frac{1}{2} & - \frac{2}{5}\\0 & 0 & \frac{1}{5}\end{matrix}\right]\).
9. Multiplication verifies AA⁻¹=A⁻¹A=I.