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## Homework Statement

Show that the matrix ##A## is of full rank if and only if ##ad-bc \neq 0## where $$A = \begin{bmatrix}

a & b \\

b & c

\end{bmatrix}$$

## Homework Equations

## The Attempt at a Solution

Suppose that the matrix ##A## is of full rank. That is, rank ##2##. Then by the rank-nullity theorem, the

dimension of the kernel is ##0##. This implies that there exists an inverse ##A^{-1}## but this will only occur if ##ad-bc \neq 0## otherwise our matrix ##A## will be singular. On the other hand, suppose ##ad-bc \neq 0##. Hence, ##A## is nonsingular and there exists an inverse ##A^{-1}## but this will occur only when the dimension of the kernel is ##0##, that is, of rank ##n = 2##.