Let `A=[[2,2],[9,4]]` and `I=[[1,0],[0,1]]` then value of `10 A^(-1)` is-

Options

A. `4I-A`
B. `6I-A`
C. `A-4I`
D. `A-6I`

Explanation

To find the value of \(10 A^{-1}\) for the matrix \(A = \begin{bmatrix} 2 & 2 \\ 9 & 4 \end{bmatrix}\), we will follow these steps:

### Step 1: Calculate the Determinant of \(A\)
The determinant of a 2x2 matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by:
\[ \text{det}(A) = ad - bc \]
For our matrix \(A\):
\[ \text{det}(A) = (2)(4) - (2)(9) = 8 - 18 = -10 \]

### Step 2: Calculate the Adjoint of \(A\)
The adjoint of a 2x2 matrix \(A\) is given by:
\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
For our matrix \(A\):
\[ \text{adj}(A) = \begin{bmatrix} 4 & -2 \\ -9 & 2 \end{bmatrix} \]

### Step 3: Calculate the Inverse of \(A\)
The inverse of a matrix \(A\) can be calculated using the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{-10} \cdot \begin{bmatrix} 4 & -2 \\ -9 & 2 \end{bmatrix} = \begin{bmatrix} -\frac{4}{10} & \frac{2}{10} \\ \frac{9}{10} & -\frac{2}{10} \end{bmatrix} = \begin{bmatrix} -0.4 & 0.2 \\ 0.9 & -0.2 \end{bmatrix} \]

### Step 4: Calculate \(10 A^{-1}\)
Now we multiply \(A^{-1}\) by 10:
\[ 10 A^{-1} = 10 \cdot \begin{bmatrix} -0.4 & 0.2 \\ 0.9 & -0.2 \end{bmatrix} = \begin{bmatrix} -4 & 2 \\ 9 & -2 \end{bmatrix} \]

### Final Answer
Thus, the value of \(10 A^{-1}\) is:
\[ \begin{bmatrix} -4 & 2 \\ 9 & -2 \end{bmatrix} \]

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