Question

If \[A = \left[ {\begin{array}{*{20}{c}}2&3\\4&6\end{array}} \right]\], then find the value of \[{A^{ - 1}}\].
A. \[\left[ {\begin{array}{*{20}{c}}1&2\\{\dfrac{{ - 3}}{2}}&3\end{array}} \right]\]
B. \[\left[ {\begin{array}{*{20}{c}}2&{ - 3}\\4&6\end{array}} \right]\]
C. \[\left[ {\begin{array}{*{20}{c}}{ - 2}&4\\{ - 3}&6\end{array}} \right]\]
D. Does not exist

Answer 1

Hint: A \[2 \times 2\] matrix is given. First, calculate the determinant of the matrix \[A\]. If the value of the determinant is non-zero. Then calculate the adjoint matrix of the given matrix and substitute the values in the formula for the inverse matrix \[{A^{ - 1}} = \dfrac{1}{{\left| A \right|}}adj\left( A \right)\] to get the required answer. If the value of the determinant is zero, then declare that the inverse matrix does not exist.

Formula used:
The adjoint matrix of a \[2 \times 2\] matrix \[A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\] is: \[adj A = \left[ {\begin{array}{*{20}{c}}d&{ - b}\\{ - c}&a\end{array}} \right]\]
The determinant of a \[2 \times 2\] matrix \[A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\] is: \[\left| A \right| = ad - bc\]
The inverse matrix: \[{A^{ - 1}} = \dfrac{1}{{\left| A \right|}}adj\left( A \right)\]

Complete step by step solution:
The given matrix is \[A = \left[ {\begin{array}{*{20}{c}}2&3\\4&6\end{array}} \right]\].

Let’s calculate the determinant of the given matrix \[A\].
Apply the formula for the determinant of a \[2 \times 2\] matrix.
We get,
\[\left| A \right| = 2 \times 6 - 4 \times 3\]
\[ \Rightarrow \left| A \right| = 12 - 12\]
\[ \Rightarrow \left| A \right| = 0\]
Since the value of the determinant is zero. So, the inverse matrix for the given matrix \[A\] does not exist.
Hence the correct option is D.

Note: Students should keep in mind that iIf the determinant of a matrix is 0, then \[\dfrac{1}{{det A}}\] is undefined. So, the matrix with a 0 determinant has no inverse. And while calculating the inverse matrix, first check whether the determinant is nonzero or not.