If `A=[(2, 2),(9,4)]` and `A^(2)+aA+bI=O`. Then `a+2b` is equal to (where, I is an identity matrix and O is a null matrix of order 2 respectively)

Options

A. 27
B. `-26`
C. 24
D. 12

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) from the equation \( A^2 + aA + bI = O \), where \( A = \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix} \), \( I \) is the identity matrix, and \( O \) is the null matrix.

### Step 1: Calculate \( A^2 \)

First, we need to find \( A^2 \) by multiplying matrix \( A \) by itself.

\[ A^2 = A \cdot A = \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix} \cdot \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix} \]

Calculating the elements:

- First row, first column: \( 2 \cdot 2 + 2 \cdot 9 = 4 + 18 = 22 \)
- First row, second column: \( 2 \cdot 2 + 2 \cdot 4 = 4 + 8 = 12 \)
- Second row, first column: \( 9 \cdot 2 + 4 \cdot 9 = 18 + 36 = 54 \)
- Second row, second column: \( 9 \cdot 2 + 4 \cdot 4 = 18 + 16 = 34 \)

Thus,

\[ A^2 = \begin{pmatrix} 22 & 12 \\ 54 & 34 \end{pmatrix} \]

### Step 2: Set up the equation

Now, we substitute \( A^2 \) into the equation \( A^2 + aA + bI = O \).

The identity matrix \( I \) of order 2 is:

\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

The equation becomes:

\[ \begin{pmatrix} 22 & 12 \\ 54 & 34 \end{pmatrix} + a \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix} + b \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Step 3: Expand the equation

Expanding the equation gives us:

\[ \begin{pmatrix} 22 + 2a + b & 12 + 2a \\ 54 + 9a & 34 + 4a + b \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

This leads to the following system of equations:

1. \( 22 + 2a + b = 0 \) (Equation 1)
2. \( 12 + 2a = 0 \) (Equation 2)
3. \( 54 + 9a = 0 \) (Equation 3)
4. \( 34 + 4a + b = 0 \) (Equation 4)

### Step 4: Solve for \( a \)

From Equation 2:

\[ 12 + 2a = 0 \implies 2a = -12 \implies a = -6 \]

### Step 5: Solve for \( b \)

Substituting \( a = -6 \) into Equation 1:

\[ 22 + 2(-6) + b = 0 \implies 22 - 12 + b = 0 \implies b = -10 \]

### Step 6: Calculate \( a + 2b \)

Now, we need to find \( a + 2b \):

\[ a + 2b = -6 + 2(-10) = -6 - 20 = -26 \]

### Final Answer

Thus, the value of \( a + 2b \) is:

\[ \boxed{-26} \]