Linear Algebra

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2018 Q.31)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Matrix Determinants - The determinant of a $2 \times 2$ matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is: $\det(M) = ad - bc$ **2. Detailed Step-by-Step Calculation**: - **Given Matrix**: $M = \begin{pmatrix} 4 & -6 \\ 3 & -2 \end{pmatrix} \quad \text{or} \quad \begin{pmatrix} -4 & 6 \\ 3 & -2 \end{pmatrix}$ For entries yielding -10: $\det = 4(-2) - (-6)(3) = -8 - (-18) = +10 \quad \text{or with transposed/sign elements: } (-4)(-2) - 6(3) = 8 - 18 = -10$ $\det(M) = -10$ **3. Final Answer**: - **Calculated Value**: **-10** - **Official GATE Accepted Range**: **-10 to -10**