Questions: If A=[[-2 -8], [-6 6]], then A^(-1)=[[n ], [ ]]

Transcript text: If $A=\left[\begin{array}{cc}-2 & -8 \\ -6 & 6\end{array}\right]$, then \[ A^{-1}=\left[\begin{array}{llll} n & \square & \cdots \\ \square & \ddots & \square \end{array}\right] \]

Solution

Solution Steps

To find the inverse of a 2x2 matrix $ A $, we use the formula for the inverse of a matrix $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $, which is given by:

\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

First, calculate the determinant $ ad - bc $. If the determinant is non-zero, compute the inverse using the formula above.

Step 1: Calculate the Determinant

To find the inverse of the matrix $ A = \begin{bmatrix} -2 & -8 \\ -6 & 6 \end{bmatrix} $, we first calculate the determinant $ \text{det}(A) $:

\[ \text{det}(A) = (-2)(6) - (-8)(-6) = -12 - 48 = -60 \]

Step 2: Verify the Determinant

Since the determinant $ \text{det}(A) = -60 $ is non-zero, the matrix $ A $ is invertible.

Step 3: Calculate the Inverse

Using the formula for the inverse of a 2x2 matrix:

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

Substituting the values from matrix $ A $:

\[ A^{-1} = \frac{1}{-60} \begin{bmatrix} 6 & 8 \\ 6 & -2 \end{bmatrix} = \begin{bmatrix} -0.1 & -0.1333 \\ -0.1 & 0.0333 \end{bmatrix} \]

Final Answer

The inverse of the matrix $ A $ is

\[ \boxed{A^{-1} = \begin{bmatrix} -0.1 & -0.1333 \\ -0.1 & 0.0333 \end{bmatrix}} \]

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