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.