NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.5 Question 10

✅ Step-by-Step Solution

1. For a 2×2 matrix use det A=a₁₁a₂₂−a₁₂a₂₁.
2. \(10\) − \(6\) = \(4\).
3. First check whether this determinant is zero.
4. The cofactor matrix is \(\left[\begin{matrix}2 & -3\\-2 & 5\end{matrix}\right]\).
5. Transpose the cofactor matrix: adj A=\(\left[\begin{matrix}2 & -2\\-3 & 5\end{matrix}\right]\).
6. Divide by det A: A⁻¹=\(\left[\begin{matrix}\frac{1}{2} & - \frac{1}{2}\\- \frac{3}{4} & \frac{5}{4}\end{matrix}\right]\).
7. Multiplication verifies AA⁻¹=A⁻¹A=I.
8. Multiply X=A⁻¹B=\(\left[\begin{matrix}-1\\4\end{matrix}\right]\).
9. Substituting this vector into the original equations gives AX=\(\left[\begin{matrix}3\\5\end{matrix}\right]\)=B. Hence the system has a unique solution.