GATE BT 2016 • Q64 NAT • 2 Marks

General Aptitude

Quantitative Aptitude

Subtopic: Data Interpretation

Question Statement:

The binary operation □ is defined as a □ b = ab+(a+b), where a and b are any two real numbers. The value of the identity element of this operation, defined as the number x such that a □ x = a, for any a, is .

Official Answer Key:

160.0

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2016 Q.64)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Triangular Matrix Determinants - For an upper-triangular or lower-triangular matrix, the determinant is simply the product of the main diagonal entries: $\det(A) = \prod_{i=1}^n a_{ii}$ **2. Detailed Step-by-Step Calculation**: - **Given Upper-Triangular Matrix**: $A = \begin{pmatrix} 5 & 16 & 81 \\ 0 & 2 & 2 \\ 0 & 0 & 16 \end{pmatrix}$ - **Diagonal Entries**: $a_{11} = 5, \quad a_{22} = 2, \quad a_{33} = 16$ - **Determinant**: $\det(A) = 5 \times 2 \times 16 = 160.0$ **3. Final Answer**: - **Calculated Value**: **160.0** - **Official GATE Accepted Range**: **160.0 to 160.0**

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