# Find the inverse of the matrix (if it exists) $A = \left( {\begin{array}{*{20}{c}}

2&{ - 2} \\

4&3

\end{array}} \right)$

Answer

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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.

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.

Last updated date: 01st Oct 2023

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