Question

Find the inverse of the matrix (if it exists) $A = \left( {\begin{array}{*{20}{c}}
  2&{ - 2} \\
  4&3
\end{array}} \right)$

Answer 1

Hint: There are two ways to determine whether the inverse of a square matrix exists.
i) Determine its rank. The rank of a matrix is a unique number associated with a square matrix. If the rank of an n x n matrix is less than n, the matrix does not have an inverse.
ii) Compute its determinant. The determinant is another unique number associated with a square matrix. When the determinant for a square matrix is equal to zero, the inverse for that matrix does not exist.

We know that if we have a matrix $X$ $ = $ $\left( {\begin{array}{*{20}{c}}
  {{a_{}}}&b \\
  c&d
\end{array}} \right)$
Inverse of $X = \dfrac{1}{{ad - bc}}\left( {\begin{array}{*{20}{c}}
  d&{ - b} \\
  { - c}&a
\end{array}} \right)$
Therefore, if we use the above formula to find the inverse of A,
 $\left( {\begin{array}{*{20}{c}}
  2&{ - 2} \\
  4&3
\end{array}} \right) = \dfrac{1}{{14}}\left( {\begin{array}{*{20}{c}}
  3&2 \\
  { - 4}&2
\end{array}} \right)$

Note: Make sure to take the signs right. Alternatively the inverse of a matrix can be found by using row or column operations.