If matrix `A=[(3,-3),(-3,3)]` and `A^2=lamdaA`, then find the value of `lamda`

Options

A. `-4`
B. 5
C. 6
D. 2

Answer

To solve the problem, we need to find the value of \(\lambda\) given that \(A^2 = \lambda A\) for the matrix \(A = \begin{pmatrix} 3 & -3 \\ -3 & 3 \end{pmatrix}\).

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

To find \(A^2\), we multiply matrix \(A\) by itself:

\[ A^2 = A \cdot A = \begin{pmatrix} 3 & -3 \\ -3 & 3 \end{pmatrix} \cdot \begin{pmatrix} 3 & -3 \\ -3 & 3 \end{pmatrix} \]

### Step 2: Perform the matrix multiplication

We will calculate each element of the resulting matrix:

- The element at position (1,1):
\[ 3 \cdot 3 + (-3) \cdot (-3) = 9 + 9 = 18 \]

- The element at position (1,2):
\[ 3 \cdot (-3) + (-3) \cdot 3 = -9 - 9 = -18 \]

- The element at position (2,1):
\[ -3 \cdot 3 + 3 \cdot (-3) = -9 - 9 = -18 \]

- The element at position (2,2):
\[ -3 \cdot (-3) + 3 \cdot 3 = 9 + 9 = 18 \]

Putting these together, we have:

\[ A^2 = \begin{pmatrix} 18 & -18 \\ -18 & 18 \end{pmatrix} \]

### Step 3: Set up the equation \(A^2 = \lambda A\)

Now we know \(A^2\), we set up the equation:

\[ \begin{pmatrix} 18 & -18 \\ -18 & 18 \end{pmatrix} = \lambda \begin{pmatrix} 3 & -3 \\ -3 & 3 \end{pmatrix} \]

### Step 4: Express the right-hand side

The right-hand side can be expressed as:

\[ \lambda A = \begin{pmatrix} 3\lambda & -3\lambda \\ -3\lambda & 3\lambda \end{pmatrix} \]

### Step 5: Equate corresponding elements

Now we equate the corresponding elements from both matrices:

1. From the (1,1) position:
\[ 18 = 3\lambda \implies \lambda = \frac{18}{3} = 6 \]

2. From the (1,2) position:
\[ -18 = -3\lambda \implies \lambda = \frac{18}{3} = 6 \]

3. From the (2,1) position:
\[ -18 = -3\lambda \implies \lambda = \frac{18}{3} = 6 \]

4. From the (2,2) position:
\[ 18 = 3\lambda \implies \lambda = \frac{18}{3} = 6 \]

### Conclusion

In all cases, we find that \(\lambda = 6\).

Thus, the value of \(\lambda\) is:

\[ \boxed{6} \]