The number of all possible matrices of order \( 2\times 2 \) with each entry 0 or 1 having their determinants as positive value is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 3
Let the matrix be \( \begin{bmatrix}a & b\\c & d\end{bmatrix} \) with each entry 0 or 1, so
\( \displaystyle |A|=ad-bc \)
Each of ad and bc is 0 or 1. The determinant is positive only when ad = 1 and bc = 0.
\( \displaystyle ad=1\ \Rightarrow\ a=d=1\ \ (1\text{ way}) \)
\( \displaystyle bc=0\ \Rightarrow\ (b,c)=(0,0),\ (0,1),\ (1,0)\ \ (3\text{ ways}) \)
Number of matrices = 1 × 3 = 3:
\( \displaystyle \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix},\ \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix},\ \begin{bmatrix}1 & 0\\1 & 1\end{bmatrix} \)