NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.4 Question 9

✅ Step-by-Step Solution

1. Expand the determinant along row 1, using cofactors Cᵢⱼ=(−1)ⁱ⁺ʲMᵢⱼ.
2. The cofactors in this row are \(\left[\begin{matrix}-1 & -4 & 1\end{matrix}\right]\).
3. The expansion terms are \(-2\), \(-4\), \(3\).
4. Adding and simplifying gives det A = \(-3\).
5. First check whether this determinant is zero.
6. The cofactor matrix is \(\left[\begin{matrix}-1 & -4 & 1\\5 & 23 & -11\\3 & 12 & -6\end{matrix}\right]\).
7. Transpose the cofactor matrix: adj A=\(\left[\begin{matrix}-1 & 5 & 3\\-4 & 23 & 12\\1 & -11 & -6\end{matrix}\right]\).
8. Divide by det A: A⁻¹=\(\left[\begin{matrix}\frac{1}{3} & - \frac{5}{3} & -1\\\frac{4}{3} & - \frac{23}{3} & -4\\- \frac{1}{3} & \frac{11}{3} & 2\end{matrix}\right]\).
9. Multiplication verifies AA⁻¹=A⁻¹A=I.