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To find the inverse of the matrix [tex]\( A = \begin{pmatrix} 2 & 3 \\ 4 & -5 \end{pmatrix} \)[/tex], we need to follow these steps:
### Step 1: Calculate the Determinant of [tex]\( A \)[/tex]
The determinant of a 2x2 matrix [tex]\( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex] is given by:
[tex]\[ \text{det}(A) = ad - bc \][/tex]
For the given matrix [tex]\( A \)[/tex]:
[tex]\[ \text{det}(A) = (2 \cdot -5) - (3 \cdot 4) = -10 - 12 = -22 \][/tex]
### Step 2: Calculate the Cofactor Matrix
The cofactor matrix is derived by finding the minors of each element and then applying a sign change pattern based on their position.
The minors for each element are:
- Minor of 2 (top-left): determinant of the submatrix formed by removing the 1st row and 1st column, which is [tex]\( -5 \)[/tex]
- Minor of 3 (top-right): determinant of the submatrix formed by removing the 1st row and 2nd column, which is [tex]\( 4 \)[/tex]
- Minor of 4 (bottom-left): determinant of the submatrix formed by removing the 2nd row and 1st column, which is [tex]\( 3 \)[/tex]
- Minor of -5 (bottom-right): determinant of the submatrix formed by removing the 2nd row and 2nd column, which is [tex]\( 2 \)[/tex]
Now apply the sign changes:
[tex]\[ \begin{pmatrix} -5 & -4 \\ -3 & 2 \end{pmatrix} \][/tex]
### Step 3: Find the Adjugate Matrix
The adjugate matrix (or adjoint matrix) is the transpose of the cofactor matrix. So, we take the transpose of the cofactor matrix:
[tex]\[ \text{adj}(A) = \begin{pmatrix} -5 & -4 \\ -3 & 2 \end{pmatrix}^T = \begin{pmatrix} -5 & -3 \\ -4 & 2 \end{pmatrix} \][/tex]
### Step 4: Calculate the Inverse of [tex]\( A \)[/tex]
The inverse of the matrix [tex]\( A \)[/tex] is given by:
[tex]\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \][/tex]
Substitute the values we found:
[tex]\[ A^{-1} = \frac{1}{-22} \begin{pmatrix} -5 & -3 \\ -4 & 2 \end{pmatrix} \][/tex]
Multiply each element of the adjugate matrix by [tex]\( \frac{1}{-22} \)[/tex]:
[tex]\[ A^{-1} = \begin{pmatrix} \frac{-5}{-22} & \frac{-3}{-22} \\ \frac{-4}{-22} & \frac{2}{-22} \end{pmatrix} = \begin{pmatrix} \frac{5}{22} & \frac{3}{22} \\ \frac{4}{22} & \frac{-2}{22} \end{pmatrix} = \begin{pmatrix} 0.22727273 & 0.13636364 \\ 0.18181818 & -0.09090909 \end{pmatrix} \][/tex]
So, the inverse of [tex]\( A \)[/tex] is:
[tex]\[ A^{-1} = \begin{pmatrix} 0.22727273 & 0.13636364 \\ 0.18181818 & -0.09090909 \end{pmatrix} \][/tex]
### Step 1: Calculate the Determinant of [tex]\( A \)[/tex]
The determinant of a 2x2 matrix [tex]\( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex] is given by:
[tex]\[ \text{det}(A) = ad - bc \][/tex]
For the given matrix [tex]\( A \)[/tex]:
[tex]\[ \text{det}(A) = (2 \cdot -5) - (3 \cdot 4) = -10 - 12 = -22 \][/tex]
### Step 2: Calculate the Cofactor Matrix
The cofactor matrix is derived by finding the minors of each element and then applying a sign change pattern based on their position.
The minors for each element are:
- Minor of 2 (top-left): determinant of the submatrix formed by removing the 1st row and 1st column, which is [tex]\( -5 \)[/tex]
- Minor of 3 (top-right): determinant of the submatrix formed by removing the 1st row and 2nd column, which is [tex]\( 4 \)[/tex]
- Minor of 4 (bottom-left): determinant of the submatrix formed by removing the 2nd row and 1st column, which is [tex]\( 3 \)[/tex]
- Minor of -5 (bottom-right): determinant of the submatrix formed by removing the 2nd row and 2nd column, which is [tex]\( 2 \)[/tex]
Now apply the sign changes:
[tex]\[ \begin{pmatrix} -5 & -4 \\ -3 & 2 \end{pmatrix} \][/tex]
### Step 3: Find the Adjugate Matrix
The adjugate matrix (or adjoint matrix) is the transpose of the cofactor matrix. So, we take the transpose of the cofactor matrix:
[tex]\[ \text{adj}(A) = \begin{pmatrix} -5 & -4 \\ -3 & 2 \end{pmatrix}^T = \begin{pmatrix} -5 & -3 \\ -4 & 2 \end{pmatrix} \][/tex]
### Step 4: Calculate the Inverse of [tex]\( A \)[/tex]
The inverse of the matrix [tex]\( A \)[/tex] is given by:
[tex]\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \][/tex]
Substitute the values we found:
[tex]\[ A^{-1} = \frac{1}{-22} \begin{pmatrix} -5 & -3 \\ -4 & 2 \end{pmatrix} \][/tex]
Multiply each element of the adjugate matrix by [tex]\( \frac{1}{-22} \)[/tex]:
[tex]\[ A^{-1} = \begin{pmatrix} \frac{-5}{-22} & \frac{-3}{-22} \\ \frac{-4}{-22} & \frac{2}{-22} \end{pmatrix} = \begin{pmatrix} \frac{5}{22} & \frac{3}{22} \\ \frac{4}{22} & \frac{-2}{22} \end{pmatrix} = \begin{pmatrix} 0.22727273 & 0.13636364 \\ 0.18181818 & -0.09090909 \end{pmatrix} \][/tex]
So, the inverse of [tex]\( A \)[/tex] is:
[tex]\[ A^{-1} = \begin{pmatrix} 0.22727273 & 0.13636364 \\ 0.18181818 & -0.09090909 \end{pmatrix} \][/tex]
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