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

✅ Step-by-Step Solution

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