Let $$A$$ be a $$\,2 \times 2$$ matrix with non-zero entries and let $${A^2} = I,$$
where $$I$$ is $$2 \times 2$$ identity matrix. Define
$$Tr$$$$(A)=$$ sum of diagonal elements of $$A$$ and $$\left| A \right| = $$ determinant of matrix $$A$$.
Statement- 1: $$Tr$$$$(A)=0$$.
Statement- 2: $$\left| A \right| = 1$$ .

Consider the system of linear equations; $$$\matrix{ {{x_1} + 2{x_2} + {x_3} = 3} \cr {2{x_1} + 3{x_2} + {x_3} = 3} \cr {3{x_1} + 5{x_2} + 2{x_3} = 1} \cr } $$$
The system has :

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

Let $$A$$ be a $$\,2 \times 2$$ matrix
Statement - 1 : $$adj\left( {adj\,A} \right) = A$$
Statement - 2 :$$\left| {adj\,A} \right| = \left| A \right|$$