⎡​1−*λ*00​22−*λ*0​332−*λ*​⎦⎤​=⎣⎡​−100​200​330​⎦⎤​


The matrix equation you provided appears to involve finding the values of \(\lambda\) (likely the eigenvalue) for which the matrix on the left-hand side equals the matrix on the right-hand side.

The equation is:

\[
\begin{bmatrix}
1 - \lambda & 0 & 0 \\
2 & 2 - \lambda & 0 \\
3 & 3 & 2 - \lambda
\end{bmatrix}
=
\begin{bmatrix}
-1 & 0 & 0 \\
2 & 0 & 0 \\
3 & 3 & 0
\end{bmatrix}
\]

To solve for \(\lambda\), we need to compare each entry of the two matrices.

### Step 1: Compare the first row
\[
1 - \lambda = -1
\]
Solving for \(\lambda\):
\[
\lambda = 2
\]

### Step 2: Compare the second row
The second row comparison gives us two equations:
1. \(2 = 2\) (This is always true, so no information is obtained from this.)
2. \(2 - \lambda = 0\)

Solving for \(\lambda\):
\[
\lambda = 2
\]

### Step 3: Compare the third row
Similarly, the third row comparison gives us two more equations:
1. \(3 = 3\) (This is always true.)
2. \(2 - \lambda = 0\)

Again, solving for \(\lambda\):
\[
\lambda = 2
\]

### Conclusion:
The value of \(\lambda\) that satisfies the matrix equation is \(\lambda = 2\).

Would you like to explore eigenvalues and eigenvectors further? Here are five related questions:

1. Find the eigenvalues of \(\begin{bmatrix} 4 & 0 \\ 1 & 3 \end{bmatrix}\).
2. Solve for \(\lambda\) in \(\begin{bmatrix} 5 - \lambda & 0 \\ 0 & 2 - \lambda \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\).
3. Find the characteristic equation of \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\).
4. Find the eigenvalues of a diagonal matrix.
5. Determine the eigenvectors of \(\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}\).

**Tip:** In diagonal matrices, the eigenvalues are the diagonal elements themselves.

⎡​1−λ 0 0​⎤​⎡2−λ 0​⎤⎡3 3 2−λ​⎦ = ⎣⎡−1 0 0⎤​⎡2 0​⎤⎡3 3 0⎦

Learn how to solve a matrix equation by finding the eigenvalue λ. This problem involves matrix comparison, eigenvalue equations, and applying linear algebra techniques. Step-by-step solution suitable for Grades 11-12 or early college students.

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