NIMCET 2024 — Mathematics PYQ

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Question

NIMCET 2024 — Mathematics PYQ

NIMCET | Mathematics | 2024

For an invertible matrix $A$ , which of the following is not always true:

Choose the correct answer:

Explanation

Solution:

\text{Consider the property: } A \cdot \text{adj}(A) = |A|I

\text{Taking determinant on both sides:}

|A \cdot \text{adj}(A)| = ||A|I|

\text{For a matrix of order } n:

||A|I| = |A|^n |I| = |A|^n

\text{Thus, } |A \cdot \text{adj}(A)| = |A|^n

\text{If } |A| = 1, \text{ then } |A \cdot \text{adj}(A)| = 1^n = 1.

\text{In this specific case, the statement } |A \cdot \text{adj}(A)| \neq 1 \text{ becomes false.}

\text{Since an invertible matrix can have a determinant equal to } 1 \text{ (e.g., identity matrix), the condition } |A \cdot \text{adj}(A)| \neq 1 \text{ is not always true.}