NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.5 Question 16

✅ Step-by-Step Solution

1. Expand the determinant along row 1, using cofactors Cᵢⱼ=(−1)ⁱ⁺ʲMᵢⱼ.
2. The cofactors in this row are \(\left[\begin{matrix}0 & 30 & -20\end{matrix}\right]\).
3. The expansion terms are \(0\), \(90\), \(-40\).
4. Adding and simplifying gives det A = \(50\).
5. First check whether this determinant is zero.
6. The cofactor matrix is \(\left[\begin{matrix}0 & 30 & -20\\-5 & 0 & 10\\10 & -20 & 10\end{matrix}\right]\).
7. Transpose the cofactor matrix: adj A=\(\left[\begin{matrix}0 & -5 & 10\\30 & 0 & -20\\-20 & 10 & 10\end{matrix}\right]\).
8. Divide by det A: A⁻¹=\(\left[\begin{matrix}0 & - \frac{1}{10} & \frac{1}{5}\\\frac{3}{5} & 0 & - \frac{2}{5}\\- \frac{2}{5} & \frac{1}{5} & \frac{1}{5}\end{matrix}\right]\).
9. Multiplication verifies AA⁻¹=A⁻¹A=I.
10. Multiply X=A⁻¹B=\(\left[\begin{matrix}5\\8\\8\end{matrix}\right]\).
11. Substituting this vector into the original equations gives AX=\(\left[\begin{matrix}60\\90\\70\end{matrix}\right]\)=B. Hence the system has a unique solution.