Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
Let $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 5$ and $\Delta' = \begin{vmatrix} b_2c_3 - b_3c_2 & a_2c_3 - a_3c_2 & a_2b_3 - a_3b_2 \\ b_3c_1 - b_1c_3 & a_3c_1 - a_1c_3 & a_3b_1 - a_1b_3 \\ b_1c_2 - b_2c_1 & a_1c_2 - a_2c_1 & a_1b_2 - a_2b_1 \end{vmatrix}$. Find $\Delta'$.
Step-by-Step Solution
Key Concept: For an $n \times n$ matrix, $\det(\text{adj}(A)) = (\det(A))^{n-1}$.
The determinant $\Delta'$ consists of all $2 \times 2$ minors of $\Delta$. Specifically, each entry in $\Delta'$ is a cofactor of the corresponding position in the original matrix. The determinant $\Delta'$ is the determinant of the adjugate (or adjoint) matrix of $\Delta$. For a $3 \times 3$ matrix, if $\text{adj}(\Delta)$ denotes the adjugate, then $\det(\text{adj}(\Delta)) = (\det(\Delta))^{n-1} = (\Delta)^{3-1} = (\Delta)^2 = 5^2 = 25$.
Correct Answer: 25