Question

How do you find the inverse of $A=\left[ \begin{matrix} 3 & 5 \\\ 2 & 4 \\\ \end{matrix} \right]$ ?

Answer 1

For finding the inverse matrix of the given matrix, first of all we will check that if it is possible to find the inverse matrix or not by calculating its determinants. If the determinant is not equal to zero that means we can get the inverse of the given matrix. Now, we will calculate the adjoint matrix of the given matrix. After that we will use the formula of inverse matrix that is:
$\Rightarrow {{A}^{-1}}=\dfrac{1}{\left| A \right|}adj\text{ A}$

Complete step-by-step answer:
Since, we have a matrix in the question that is:
$\Rightarrow A=\left[ \begin{matrix} 3 & 5 \\\ 2 & 4 \\\ \end{matrix} \right]$
Now, we will calculate its determinants as:
$\Rightarrow 3\times 4-2\times 5$
Here, we will complete the multiplication first as:
$\Rightarrow 12-10$
Now, we will subtract $10$ from $12$ and will have:
$\Rightarrow 2$
Since, determinants of a given matrix are not equal to zero that means we will get the inverse of the given matrix. Now, we will find the matrix of the cofactor of element of given matrix as:
$\Rightarrow A'=\left[ \begin{matrix} 4 & -2 \\\ -5 & 3 \\\ \end{matrix} \right]$
Now, we will calculate the adjoint of the given matrix as:
$\Rightarrow adj\text{ A = transpose matrix of A }\\!\\!'\\!\\!\text{ }$
$\Rightarrow adj\text{ A = }\left[ \begin{matrix} 4 & -5 \\\ -2 & 3 \\\ \end{matrix} \right]$
Since, we got the determinant of given matrix and adjoint matrix of the given matrix. So, we will calculate the inverse matrix of the given matrix as:
$\Rightarrow {{A}^{-1}}=\dfrac{1}{\left| A \right|}adj\text{ A}$
Now, we will apply the value of determinant and adjoint matrix in the above formula as:
$\Rightarrow {{A}^{-1}}=\dfrac{1}{2}\text{ }\left[ \begin{matrix} 4 & -5 \\\ -2 & 3 \\\ \end{matrix} \right]$
Here, we will multiply the outer value with each element of the adjoint matrix as:
$\Rightarrow {{A}^{-1}}=\text{ }\left[ \begin{matrix} \dfrac{1}{2}\times 4 & \dfrac{1}{2}\times -5 \\\ \dfrac{1}{2}\times -2 & \dfrac{1}{2}\times 3 \\\ \end{matrix} \right]$
Now, we will the required calculation to complete the process as:
$\Rightarrow {{A}^{-1}}=\text{ }\left[ \begin{matrix} 2 & \dfrac{-5}{2} \\\ -1 & \dfrac{3}{2} \\\ \end{matrix} \right]$
Hence, the inverse matrix of the given matrix is $\left[ \begin{matrix} 2 & \dfrac{-5}{2} \\\ -1 & \dfrac{3}{2} \\\ \end{matrix} \right]$ .

Note: Here, we need to remember some points to find the inverse of any matrix. First we need to calculate the determinant of the given matrix. If it is not equal to zero that means we will get the inverse of the given matrix, otherwise not. Then, we need to remember to calculate the matrix of cofactor of elements of given matrix and it will be as:
Let, the matrix is $A$ that is:
$A=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right]$
Now, the matrix of cofactor of the above matrix will be:
$B=\left[ \begin{matrix} {{a}_{22}} & -{{a}_{21}} \\\ -{{a}_{12}} & {{a}_{11}} \\\ \end{matrix} \right]$
Then, we will calculate transpose matrix of $B$ that is adjoint matrix of $A$ as:
$adj\text{ A}=\left[ \begin{matrix} {{a}_{22}} & -{{a}_{12}} \\\ -{{a}_{21}} & {{a}_{11}} \\\ \end{matrix} \right]$
After that we will use the formula for getting the inverse matrix.