NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.5 Question 15

✅ Step-by-Step Solution

1. Expand the determinant along row 1, using cofactors Cᵢⱼ=(−1)ⁱ⁺ʲMᵢⱼ.
2. The cofactors in this row are \(\left[\begin{matrix}0 & 2 & 1\end{matrix}\right]\).
3. The expansion terms are \(0\), \(-6\), \(5\).
4. Adding and simplifying gives det A = \(-1\).
5. First check whether this determinant is zero.
6. The cofactor matrix is \(\left[\begin{matrix}0 & 2 & 1\\-1 & -9 & -5\\2 & 23 & 13\end{matrix}\right]\).
7. Transpose the cofactor matrix: adj A=\(\left[\begin{matrix}0 & -1 & 2\\2 & -9 & 23\\1 & -5 & 13\end{matrix}\right]\).
8. Divide by det A: A⁻¹=\(\left[\begin{matrix}0 & 1 & -2\\-2 & 9 & -23\\-1 & 5 & -13\end{matrix}\right]\).
9. Multiplication verifies AA⁻¹=A⁻¹A=I.
10. Multiply X=A⁻¹B=\(\left[\begin{matrix}1\\2\\3\end{matrix}\right]\).
11. Substituting this vector into the original equations gives AX=\(\left[\begin{matrix}11\\-5\\-3\end{matrix}\right]\)=B. Hence the system has a unique solution.