Linear Algebra

### **Detailed Step-by-Step Solution (GATE AE 2018 Q.30 / AE Q.20)** **1. Intuitive Physical Setup & Given Parameters**: - Given matrix: $M = \begin{bmatrix} 1 & 1 & -1 \\ 2 & 1 & 0 \\ 3 & 1 & 1 \end{bmatrix}$ **2. Step-by-Step Mathematical Derivation**: 1. **Evaluating Determinant by Expansion along Row 2**: Using row 2 ($2, 1, 0$) due to the presence of 0: $\det(M) = -2 \begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix} + 1 \begin{vmatrix} 1 & -1 \\ 3 & 1 \end{vmatrix} - 0$ 2. **Computing $2 \times 2$ Minors**: $\begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix} = (1)(1) - (-1)(1) = 1 + 1 = 2$ $\begin{vmatrix} 1 & -1 \\ 3 & 1 \end{vmatrix} = (1)(1) - (-1)(3) = 1 + 3 = 4$ 3. **Combining**: $\det(M) = -2(2) + 1(4) = -4 + 4 = 0$ 4. **Alternative Row Operation Verification**: Notice that $\text{Row}_3 = 2\text{Row}_2 - \text{Row}_1$: $2(2, 1, 0) - (1, 1, -1) = (4-1, 2-1, 0 - (-1)) = (3, 1, 1) = \text{Row}_3$ Since Row 3 is a linear combination of Row 1 and Row 2, the rows are linearly dependent, so $\det(M) = 0$. **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Exact Evaluation**: **0.0** (Acceptable Range: [0.0, 0.0]). **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **0.0** - **Exam Rule**: Always check for linear dependence between rows before computing full cofactor expansion: if $\sum c_i R_i = 0$, $\det(A) = 0$.