Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
If $M = \begin{pmatrix} 2 & ABC^{2020} \\ D \end{pmatrix}$ (Given), $|M| = |2ABC^{2020}D|$, and $ABC^{2020}$ is a $2 \times 2$ matrix, find $|2ABC^{2020}D|$ if $|ABC^{2020}D| = 4096$.
Step-by-Step Solution
Key Concept: When a scalar multiplies all entries of an $n \times n$ matrix, the determinant is multiplied by that scalar raised to the power $n$
Given that $M = \begin{pmatrix} 2 & ABC^{2020} \\ D \end{pmatrix}$ where $ABC^{2020}$ is a $2 \times 2$ matrix. We have $|M| = 4|ABC^{2020}D|$ since the scalar 2 is applied to all entries. Therefore $|2ABC^{2020}D| = 2^2|ABC^{2020}D| = 4 \times 1024 = 4096$.
Correct Answer: 4096