Let A be a 2 × 2 real matrix with entries from \ 0, 1\ and |A| ≠ 0 . Consider the following two statements; (P) If A ≠ l _ 2 , then |A| = - 1 (Q) If |A| = 1 , then tr(A) = 2 Where l _ 2 denotes 2 × 2 identity matrix and

Correct answer: D. Both (P) and (Q) are true Let& A = [ cases a & b c & d cases ]a, b, c, d ∈ 0, 1 |A| = ad - bc ≠ 0 ⇒ ad = 1, bc = 0 ~ or ad = 0, bc = 1 (P) If A ≠ I _ 2 ⇒ad ≠ 1 ⇒ad = 0, bc = 1 ⇒ | A | = - 1 (P) is true. (Q) ~ If |A| = 1 ⇒ ad = 1, bc = 0 ⇒ tr ( A ) = 2 (Q) is true.

JEE Main 20202 SeptemberMorning Shift

Exam: JEE Main
Year: 2020
Date: 2 September
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2020 · 2 September · Morning Shift · Mathematics · Matrices

Let A be a 2 × 2 real matrix with entries from \ 0, 1\ and |A| ≠ 0...

Let $A$ be a $2 \times 2$ real matrix with entries from \left\{0, 1\right\} and \left|A\right| \ne 0 $. Consider the following two statements;\left(P\right)$ If $A \ne {l}_{2}$ , then $\left|A\right| = - 1$

\left(Q\right) $If \left|A\right| = 1$ , then $tr\left(A\right) = 2$

Where ${l}_{2}$ denotes $2 \times 2$ identity matrix and $tr\left(A\right)$ denotes the sum of the diagonal entries of $A$ . Then

Options

A $\left(P\right)$ is false and $\left(Q\right)$ is true
BBoth $\left(P\right)$ and $\left(Q\right)$ are false
C $\left(P\right)$ is true and $\left(Q\right)$ is false
DBoth $\left(P\right)$ and $\left(Q\right)$ are true

Correct answer

D. Both $\left(P\right)$ and $\left(Q\right)$ are true

Step-by-step solution

Let& $A = \left[$ \begin{cases}a & b \\ c & d\end{cases} $\right]a, b, c, d \in { 0, 1 }$

\left|A\right| = ad - bc \ne 0 $$ ⇒ ad = 1, bc = 0 $~ or$ ad = 0, bc = 1 $\left(P\right)$

If $A \ne {I}_{2} ⇒ad \ne 1 ⇒ad = 0, bc = 1 ⇒ | A | = - 1$

\left(P\right) $is true.\left(Q\right)$ ~

If \left|A\right| = 1 ⇒ ad = 1, bc = 0 ⇒ tr ( A ) = 2 $\left(Q\right)$ is true.

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