Inverse of a 2 x 2 matrix
- We can use the following formula to calculate the determinant of a 2 × 2 matrix.
\[\Large
A =
\begin{bmatrix}
\color{red}{a} & \color{green}{b} \\
\color{orange}{c} & \color{purple}{d}
\end{bmatrix}
\]
\[\Large A^{-1} = \frac{1}{det(A)} \begin{bmatrix} \color{purple}{d} & \color{green}{-b} \\ \color{orange}{-c} & \color{red}{a} \end{bmatrix} \]
\[\Large A^{-1} = \frac{1}{det(A)} \begin{bmatrix} \color{purple}{d} & \color{green}{-b} \\ \color{orange}{-c} & \color{red}{a} \end{bmatrix} \]
- Let’s find the inverse matrix of a 2 × 2 matrix.
\[\Large
A =
\begin{bmatrix}
2 & 3 \\
5 & 7 \\
\end{bmatrix}
\]
1. Find the determinant of the matrix.
- Calculate the determinant of the matrix.
- To learn how to find the determinant of a 2 × 2 matrix refer Determinant of a 2 × 2 matrix.
\[\Large
A =
\begin{bmatrix}
\color{red}{2} & \color{green}{3} \\
\color{orange}{5} & \color{purple}{7}
\end{bmatrix}
\]
\[\Large det(A) = \textcolor{red}{2} × \textcolor{purple}{7~} – \textcolor{green}{~3} × \textcolor{orange}{5} \] \[\Large det(A) = -1 \]
\[\Large det(A) = \textcolor{red}{2} × \textcolor{purple}{7~} – \textcolor{green}{~3} × \textcolor{orange}{5} \] \[\Large det(A) = -1 \]
2. Calculate the inverse matrix
- Use the given formula to calculate the inverse matrix.
- In here multiply the original matrix by 1/det(A) to calculate the inverse matrix.
- This multiplication process is known as scalar multiplication, and to learn more about scalar multiplication refer Matrix multiplication.
\[\Large
A^{-1} = \frac{1}{-1}
\begin{bmatrix}
\color{purple}{7} & \color{green}{-3} \\
\color{orange}{-5} & \color{red}{2}
\end{bmatrix}
\]
\[\Large A^{-1} = \begin{bmatrix} \color{purple}{-\frac{7}{1}} & \color{green}{\frac{3}{1}} \\ \color{orange}{\frac{5}{1}} & \color{red}{-\frac{2}{1}} \end{bmatrix} \]
\[\Large \mathbf{A^{-1} = } \begin{bmatrix} \mathbf{\color{purple}{-7}} & \mathbf{\color{green}{3}} \\ \mathbf{\color{orange}{5}} & \mathbf{\color{red}{-2}} \end{bmatrix} \]
\[\Large A^{-1} = \begin{bmatrix} \color{purple}{-\frac{7}{1}} & \color{green}{\frac{3}{1}} \\ \color{orange}{\frac{5}{1}} & \color{red}{-\frac{2}{1}} \end{bmatrix} \]
\[\Large \mathbf{A^{-1} = } \begin{bmatrix} \mathbf{\color{purple}{-7}} & \mathbf{\color{green}{3}} \\ \mathbf{\color{orange}{5}} & \mathbf{\color{red}{-2}} \end{bmatrix} \]
The following formula can be used to calculate the inverse of a 2 × 2 matrix.
