Inverse Matrix 2 by 2
Simplify Inverse Matrix 2 by 2
The 2×2 Matrix Inverse Solver is a precise tool designed to calculate the inverse of a matrix, provided it exists. It simplifies the process by performing all necessary steps and ensuring accuracy while giving a detailed explanation of the calculation.
Example 1: Inverse of a Simple Matrix
Find the inverse of:
\[
A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}
\]
Solution:
1. Calculate the Determinant:
\[
\text{det}(A) = (2 \cdot 4) – (3 \cdot 1) = 8 – 3 = 5
\]
2. Apply the Inverse Formula:
\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]
Substitute \(a = 2\), \(b = 3\), \(c = 1\), \(d = 4\):
\[
A^{-1} = \frac{1}{5} \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}
\]
3. Simplify the Matrix:
\[
A^{-1} = \begin{bmatrix} \frac{4}{5} & -\frac{3}{5} \\ -\frac{1}{5} & \frac{2}{5} \end{bmatrix}
\]
Final Answer:
\[
A^{-1} = \begin{bmatrix} 0.8 & -0.6 \\ -0.2 & 0.4 \end{bmatrix}
\]
Example 2: Inverse of a Non-Invertible Matrix
Find the inverse of:
\[
B = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}
\]
Solution:
1. Calculate the Determinant:
\[
\text{det}(B) = (1 \cdot 4) – (2 \cdot 2) = 4 – 4 = 0
\]
Since \(\text{det}(B) = 0\), the matrix \(B\) is not invertible.
Final Answer:
\[
B^{-1} \text{ does not exist.}
\]
