The inverse of the matrix [tex]\vec A = \left[\begin{array}{cc}1&2\\3&4\end{array}\right][/tex] is represented by the matrix [tex]\vec A^{-1} = \left[\begin{array}{cc}-2&1\\\frac{3}{2} &-\frac{1}{2} \end{array}\right][/tex].
How to determine the inverse matrix
A matrix A has an inverse matrix if and only if its determinant is different than 0. Given that we have a matrix formed by 2 rows and 2 columns, we can obtain the following inverse matrix by using the following formula:
[tex]\vec A ^{-1} = \frac{1}{\det (\vec A)} \cdot adj (\vec A)[/tex] (1)
Where [tex]adj (\vec A)[/tex] is the adjoint of the matrix, which is the transposed of the cofactor matrix.
If we know that [tex]\vec A = \left[\begin{array}{cc}1&2\\3&4\end{array}\right][/tex], then the inverse of the matrix is determined below:
[tex]\det (\vec A) = -2[/tex]
[tex]adj (\vec A) = \left[\begin{array}{cc}4&-2\\-3&1\end{array}\right][/tex]
[tex]\vec A^{-1} = -\frac{1}{2}\cdot \left[\begin{array}{cc}4&-2\\-3&1\end{array}\right][/tex]
[tex]\vec A^{-1} = \left[\begin{array}{cc}-2&1\\\frac{3}{2} &-\frac{1}{2} \end{array}\right][/tex]
The inverse of the matrix [tex]\vec A = \left[\begin{array}{cc}1&2\\3&4\end{array}\right][/tex] is represented by the matrix [tex]\vec A^{-1} = \left[\begin{array}{cc}-2&1\\\frac{3}{2} &-\frac{1}{2} \end{array}\right][/tex]. [tex]\blacksquare[/tex]
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