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To find the inverse of the matrix \( A \) given by:

\[
A = \begin{bmatrix} 8 & 2 \\ -7 & -3 \end{bmatrix}
\]

we use the formula for the inverse of a \( 2 \times 2 \) matrix, which is:

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]

where \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). Here, \( a = 8 \), \( b = 2 \), \( c = -7 \), and \( d = -3 \). The determinant \( \text{det}(A) \) is calculated as:

\[
\text{det}(A) = ad - bc = (8)(-3) - (2)(-7) = -24 + 14 = -10
\]

Now, we calculate \( A^{-1} \) as follows:

\[
A^{-1} = \frac{1}{-10} \begin{bmatrix} -3 & -2 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 0.3 & 0.2 \\ -0.7 & -0.8 \end{bmatrix}
\]

Thus, the inverse of matrix \( A \) is:

\[
A^{-1} = \begin{bmatrix} 0.3 & 0.2 \\ -0.7 & -0.8 \end{bmatrix}
\]

Would you like more details on this calculation or help with another question?

1. Would you like to know why the determinant is used in the inverse calculation?
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4. How to verify the correctness of the calculated matrix inverse?
5. Are you interested in learning about the properties of matrix inverses?
  
**Tip:** Always check the determinant first to ensure the matrix is invertible (non-zero determinant).

If A = [ [8, 2], [-7, -3] ], find A^{-1}.

Find the inverse of a 2x2 matrix A = [8, 2; -7, -3]. Learn step-by-step how to calculate the determinant and apply the matrix inversion formula. This problem is suitable for students in Linear Algebra courses.

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