if `A=[(1,2),(2,3)] and A^(2) -lambdaA-l_(2)=O,`then `lambda` is equal to

Options

A. `-4`
B. `-2`
C. 2
D. 4

Answer

To solve the problem, we need to find the value of \(\lambda\) given the matrix \(A\) and the equation \(A^2 - \lambda A - I_2 = O\), where \(I_2\) is the identity matrix of order 2.

### Step-by-Step Solution:

1. **Define the Matrix \(A\)**:
\[ A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} \]

2. **Calculate \(A^2\)**:
To find \(A^2\), we multiply matrix \(A\) by itself:
\[ A^2 = A \cdot A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} \]
Performing the multiplication:
- First row, first column: \(1 \cdot 1 + 2 \cdot 2 = 1 + 4 = 5\)
- First row, second column: \(1 \cdot 2 + 2 \cdot 3 = 2 + 6 = 8\)
- Second row, first column: \(2 \cdot 1 + 3 \cdot 2 = 2 + 6 = 8\)
- Second row, second column: \(2 \cdot 2 + 3 \cdot 3 = 4 + 9 = 13\)

Thus,
\[ A^2 = \begin{pmatrix} 5 & 8 \\ 8 & 13 \end{pmatrix} \]

3. **Set Up the Equation**:
We substitute \(A^2\) and \(A\) into the equation:
\[ A^2 - \lambda A - I_2 = O \]
where \(I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\).

This gives us:
\[ \begin{pmatrix} 5 & 8 \\ 8 & 13 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} - \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

4. **Combine the Matrices**:
Now, we can combine the matrices:
\[ \begin{pmatrix} 5 - \lambda & 8 - 2\lambda \\ 8 - 2\lambda & 13 - 3\lambda \end{pmatrix} - \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]
This simplifies to:
\[ \begin{pmatrix} 4 - \lambda & 8 - 2\lambda \\ 8 - 2\lambda & 12 - 3\lambda \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

5. **Set Up the Equations**:
From the above matrix equation, we can set up the following equations:
- \(4 - \lambda = 0\)
- \(8 - 2\lambda = 0\)
- \(12 - 3\lambda = 0\)

6. **Solve for \(\lambda\)**:
From the first equation:
\[ 4 - \lambda = 0 \implies \lambda = 4 \]
From the second equation:
\[ 8 - 2\lambda = 0 \implies 2\lambda = 8 \implies \lambda = 4 \]
From the third equation:
\[ 12 - 3\lambda = 0 \implies 3\lambda = 12 \implies \lambda = 4 \]

All equations give us the same result:
\[ \lambda = 4 \]

### Final Answer:
\(\lambda = 4\)