Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
If $A = \begin{vmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 1 & 4 & 3 \end{vmatrix}$, $B = \text{adj}A$ and $C = 3A$, then $\frac{|\text{adj}(B)|}{|C|}$ is equal to
Step-by-Step Solution
Key Concept: Determinant of a $3 \times 3$ matrix is calculated using cofactor expansion along any row or column.
Given the matrix $|A| = \begin{vmatrix} 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3 \end{vmatrix}$, we calculate using cofactor expansion: $|A| = 1(3 \cdot 3 - 4 \cdot (-1)) - 1(1 \cdot 3 - 4 \cdot 1) + 2(1 \cdot (-1) - 3 \cdot 1) = 1(9+4) - 1(3-4) + 2(-1-3) = 13 + 1 - 8 = 6$. The determinant value is 6, obtained through systematic cofactor expansion along the first row.
Correct Answer: 6