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  4. The number of 3 × 3 non-singular matrices with four entries as 1 and all other entries as 0 is

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Q. The number of $3 \times 3$ non-singular matrices with four entries as $1$ and all other entries as $0$ is

AIEEEAIEEE 2009Determinants

Solution:

The matrix $\begin{pmatrix}1&a&b\\ c&1&d\\ e&f&1\end{pmatrix}$ where exectly one of $a, b,c,d,e,f$ is 1 and rest of them are zero, is invertible. There are six such matrices.
Also the matrix $\begin{vmatrix}1&0&1\\ 0&1&0\\ 1&0&0\end{vmatrix}$ is invertible Thus, there are at least 7 such matrices which are invertible.