QuestionUse the fact that if $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ , then $A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right]$ to find the inverse of the given matrix, if possible. Check that $A A^{-1}=I_{2}$ and $A^{-1} A=I_{2}$ . $A=\left[\begin{array}{rr} 2 & -2 \\ -1 & 1 \end{array}\right]$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $A^{-1}=$ $\square$ (Simplify your answer.) B. The inverse matrix is not possible.

Studdy Solution

STEP 1

1. The matrix $A$ is a $2 \times 2$ matrix.
2. The formula for the inverse of a $2 \times 2$ matrix is applicable if the determinant $ad - bc \neq 0$ .
3. We need to verify that both $A A^{-1} = I_2$ and $A^{-1} A = I_2$ .

STEP 2

1. Calculate the determinant of matrix $A$ .
2. Determine if the inverse exists.
3. Calculate the inverse using the given formula.
4. Verify that $A A^{-1} = I_2$ .
5. Verify that $A^{-1} A = I_2$ .

STEP 3

Calculate the determinant of matrix $A$ :
$A = \begin{bmatrix} 2 & -2 \\ -1 & 1 \end{bmatrix}$
The determinant $\text{det}(A)$ is calculated as:
$\text{det}(A) = (2)(1) - (-2)(-1) = 2 - 2 = 0$

STEP 4

Determine if the inverse exists:
Since the determinant $\text{det}(A) = 0$ , the matrix $A$ does not have an inverse.

STEP 5

Since the determinant is zero, the inverse of matrix $A$ does not exist. Therefore, the correct choice is:
B. The inverse matrix is not possible.

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