JEE Main 2020 (Online) 2nd September Morning ShiftQuestion 1
Let $A$ be a $2\times2$ real matrix with entries from $\{0,1\}$ and $|A|\ne0$. Consider the following two statements : (P) If $A\ne I_2$, then $|A|=-1$ (Q) If $|A|=1$, then $\mathrm{tr}(A)=2$, where $I_2$ denotes $2\times2$ identity matrix and $\mathrm{tr}(A)$ denotes the sum of the diagonal entries of $A$. Then :
Matrices
Solution
Invertible 0-1 matrices: $I$, $\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $\begin{pmatrix}1&1\\1&0\end{pmatrix}$, $\begin{pmatrix}0&1\\1&1\end{pmatrix}$ (det $-1$), and $\begin{pmatrix}1&1\\0&1\end{pmatrix}$, $\begin{pmatrix}1&0\\1&1\end{pmatrix}$ (det $+1$). (P) false: the last two are $\ne I$ with det $1$. (Q) true: det $1$ matrices are $I$ and the two triangular ones, all with trace 2. Option B.