If $A=\begin{bmatrix}2&-2\ 4&3\end{bmatrix}$, then $A^{-1}=$

Q: If $A=\begin{bmatrix}2&-2\\ 4&3\end{bmatrix}$, then $A^{-1}=$

To find the inverse of the matrix $ A = \begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix} $, we use the formula for the inverse of a 2x2 matrix: If $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $, then $ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} $, provided that $ ad - bc \neq 0 $. **Step 1: Calculate the determinant of $ A $.** The determinant of $ A $ is given by: \[ \text{det}(A) = (2)(3) - (-2)(4) = 6 + 8 = 14 \] Since the determinant is not zero, the matrix $ A $ is invertible. **Step 2: Apply the formula for the inverse.** Using the formula for the inverse, we have: \[ A^{-1} = \frac{1}{14} \begin{bmatrix} 3 & 2 \\ -4 & 2 \end{bmatrix} \] This matches Option 1: $\frac{1}{14}\begin{bmatrix}3&2\\ -4&2\end{bmatrix}$. **Explanation for why other options are incorrect:** - **Option 2:** $\frac{1}{28}\begin{bmatrix}3&-4\\ 2&2\end{bmatrix}$ This option uses an incorrect scalar factor $\frac{1}{28}$ instead of $\frac{1}{14}$, and the arrangement of elements does not match the required inverse matrix. - **Option 3:** $\frac{1}{14}\begin{bmatrix}-2&2\\ -4&-3\end{bmatrix}$ This option has incorrect elements in the matrix. The positions and signs of the elements do not correspond to the correct inverse matrix. - **Option 4:** $\frac{1}{14}\begin{bmatrix}4&3\\ 2&-2\end{bmatrix}$ This option has incorrect elements in the matrix. The arrangement of elements does not match the correct inverse matrix. Thus, Option 1 is the correct answer as it accurately represents the inverse of matrix $ A $.

Question

Accepted Answer

To find the inverse of the matrix $ A = \begin{bmatrix} 2 & -2 \ 4 & 3 \end{bmatrix} $, we use the formula for the inverse of a 2x2 matrix:

If $ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} $, then $ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} $, provided that $ ad - bc \neq 0 $.

**Step 1: Calculate the determinant of $ A $.**

The determinant of $ A $ is given by:

$$
\text{det}(A) = (2)(3) - (-2)(4) = 6 + 8 = 14
$$

Since the determinant is not zero, the matrix $ A $ is invertible.

**Step 2: Apply the formula for the inverse.**

Using the formula for the inverse, we have:

$$
A^{-1} = \frac{1}{14} \begin{bmatrix} 3 & 2 \ -4 & 2 \end{bmatrix}
$$

This matches Option 1: $\frac{1}{14}\begin{bmatrix}3&2\ -4&2\end{bmatrix}$.

**Explanation for why other options are incorrect:**

- **Option 2:** $\frac{1}{28}\begin{bmatrix}3&-4\ 2&2\end{bmatrix}$

This option uses an incorrect scalar factor $\frac{1}{28}$ instead of $\frac{1}{14}$, and the arrangement of elements does not match the required inverse matrix.

- **Option 3:** $\frac{1}{14}\begin{bmatrix}-2&2\ -4&-3\end{bmatrix}$

This option has incorrect elements in the matrix. The positions and signs of the elements do not correspond to the correct inverse matrix.

- **Option 4:** $\frac{1}{14}\begin{bmatrix}4&3\ 2&-2\end{bmatrix}$

This option has incorrect elements in the matrix. The arrangement of elements does not match the correct inverse matrix.

Thus, Option 1 is the correct answer as it accurately represents the inverse of matrix $ A $.