NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.5 Question 9

✅ Step-by-Step Solution

1. For a 2×2 matrix use det A=a₁₁a₂₂−a₁₂a₂₁.
2. \(-20\) − \(-9\) = \(-11\).
3. First check whether this determinant is zero.
4. The cofactor matrix is \(\left[\begin{matrix}-5 & -3\\3 & 4\end{matrix}\right]\).
5. Transpose the cofactor matrix: adj A=\(\left[\begin{matrix}-5 & 3\\-3 & 4\end{matrix}\right]\).
6. Divide by det A: A⁻¹=\(\left[\begin{matrix}\frac{5}{11} & - \frac{3}{11}\\\frac{3}{11} & - \frac{4}{11}\end{matrix}\right]\).
7. Multiplication verifies AA⁻¹=A⁻¹A=I.
8. Multiply X=A⁻¹B=\(\left[\begin{matrix}- \frac{6}{11}\\- \frac{19}{11}\end{matrix}\right]\).
9. Substituting this vector into the original equations gives AX=\(\left[\begin{matrix}3\\7\end{matrix}\right]\)=B. Hence the system has a unique solution.