NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.5 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 \(\left[\begin{matrix}13 & 5 & 3\end{matrix}\right]\).
3. The expansion terms are \(26\), \(5\), \(3\).
4. Adding and simplifying gives det A = \(34\).
5. First check whether this determinant is zero.
6. The cofactor matrix is \(\left[\begin{matrix}13 & 5 & 3\\8 & -10 & -6\\1 & 3 & -5\end{matrix}\right]\).
7. Transpose the cofactor matrix: adj A=\(\left[\begin{matrix}13 & 8 & 1\\5 & -10 & 3\\3 & -6 & -5\end{matrix}\right]\).
8. Divide by det A: A⁻¹=\(\left[\begin{matrix}\frac{13}{34} & \frac{4}{17} & \frac{1}{34}\\\frac{5}{34} & - \frac{5}{17} & \frac{3}{34}\\\frac{3}{34} & - \frac{3}{17} & - \frac{5}{34}\end{matrix}\right]\).
9. Multiplication verifies AA⁻¹=A⁻¹A=I.
10. Multiply X=A⁻¹B=\(\left[\begin{matrix}1\\\frac{1}{2}\\- \frac{3}{2}\end{matrix}\right]\).
11. Substituting this vector into the original equations gives AX=\(\left[\begin{matrix}1\\\frac{3}{2}\\9\end{matrix}\right]\)=B. Hence the system has a unique solution.