NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.4 Question 11

✅ Step-by-Step Solution

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