G3 - Eigenvalues


Example 1 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -4 & 1 \\ 7 & 2 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -4 & 1 \\ 7 & 2 \end{array}\right] \) is \( \lambda^{2} + 2 \lambda - 15 \).

The eigenvalues of \( \left[\begin{array}{cc} -4 & 1 \\ 7 & 2 \end{array}\right] \) are \( -5 \) and \( 3 \).


Example 2 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 9 & 2 \\ -28 & -9 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 9 & 2 \\ -28 & -9 \end{array}\right] \) is \( \lambda^{2} - 25 \).

The eigenvalues of \( \left[\begin{array}{cc} 9 & 2 \\ -28 & -9 \end{array}\right] \) are \( 5 \) and \( -5 \).


Example 3 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 7 & 2 \\ -18 & -6 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 7 & 2 \\ -18 & -6 \end{array}\right] \) is \( \lambda^{2} - \lambda - 6 \).

The eigenvalues of \( \left[\begin{array}{cc} 7 & 2 \\ -18 & -6 \end{array}\right] \) are \( 3 \) and \( -2 \).


Example 4 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 1 & 2 \\ 0 & -5 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 1 & 2 \\ 0 & -5 \end{array}\right] \) is \( \lambda^{2} + 4 \lambda - 5 \).

The eigenvalues of \( \left[\begin{array}{cc} 1 & 2 \\ 0 & -5 \end{array}\right] \) are \( -5 \) and \( 1 \).


Example 5 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 6 & 1 \\ -20 & -6 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 6 & 1 \\ -20 & -6 \end{array}\right] \) is \( \lambda^{2} - 16 \).

The eigenvalues of \( \left[\begin{array}{cc} 6 & 1 \\ -20 & -6 \end{array}\right] \) are \( 4 \) and \( -4 \).


Example 6 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -1 & 1 \\ 2 & 0 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -1 & 1 \\ 2 & 0 \end{array}\right] \) is \( \lambda^{2} + \lambda - 2 \).

The eigenvalues of \( \left[\begin{array}{cc} -1 & 1 \\ 2 & 0 \end{array}\right] \) are \( -2 \) and \( 1 \).


Example 7 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 7 & 1 \\ -11 & -5 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 7 & 1 \\ -11 & -5 \end{array}\right] \) is \( \lambda^{2} - 2 \lambda - 24 \).

The eigenvalues of \( \left[\begin{array}{cc} 7 & 1 \\ -11 & -5 \end{array}\right] \) are \( 6 \) and \( -4 \).


Example 8 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 0 & 2 \\ 8 & 0 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 0 & 2 \\ 8 & 0 \end{array}\right] \) is \( \lambda^{2} - 16 \).

The eigenvalues of \( \left[\begin{array}{cc} 0 & 2 \\ 8 & 0 \end{array}\right] \) are \( -4 \) and \( 4 \).


Example 9 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 4 & 1 \\ -18 & -5 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 4 & 1 \\ -18 & -5 \end{array}\right] \) is \( \lambda^{2} + \lambda - 2 \).

The eigenvalues of \( \left[\begin{array}{cc} 4 & 1 \\ -18 & -5 \end{array}\right] \) are \( 1 \) and \( -2 \).


Example 10 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -1 & 1 \\ 21 & 3 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -1 & 1 \\ 21 & 3 \end{array}\right] \) is \( \lambda^{2} - 2 \lambda - 24 \).

The eigenvalues of \( \left[\begin{array}{cc} -1 & 1 \\ 21 & 3 \end{array}\right] \) are \( -4 \) and \( 6 \).


Example 11 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 0 & 2 \\ 4 & 2 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 0 & 2 \\ 4 & 2 \end{array}\right] \) is \( \lambda^{2} - 2 \lambda - 8 \).

The eigenvalues of \( \left[\begin{array}{cc} 0 & 2 \\ 4 & 2 \end{array}\right] \) are \( -2 \) and \( 4 \).


Example 12 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 4 & 1 \\ 2 & 5 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 4 & 1 \\ 2 & 5 \end{array}\right] \) is \( \lambda^{2} - 9 \lambda + 18 \).

The eigenvalues of \( \left[\begin{array}{cc} 4 & 1 \\ 2 & 5 \end{array}\right] \) are \( 3 \) and \( 6 \).


Example 13 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -2 & 1 \\ 4 & 1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -2 & 1 \\ 4 & 1 \end{array}\right] \) is \( \lambda^{2} + \lambda - 6 \).

The eigenvalues of \( \left[\begin{array}{cc} -2 & 1 \\ 4 & 1 \end{array}\right] \) are \( -3 \) and \( 2 \).


Example 14 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 1 & 2 \\ 12 & -1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 1 & 2 \\ 12 & -1 \end{array}\right] \) is \( \lambda^{2} - 25 \).

The eigenvalues of \( \left[\begin{array}{cc} 1 & 2 \\ 12 & -1 \end{array}\right] \) are \( -5 \) and \( 5 \).


Example 15 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -1 & 1 \\ 5 & 3 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -1 & 1 \\ 5 & 3 \end{array}\right] \) is \( \lambda^{2} - 2 \lambda - 8 \).

The eigenvalues of \( \left[\begin{array}{cc} -1 & 1 \\ 5 & 3 \end{array}\right] \) are \( -2 \) and \( 4 \).


Example 16 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -1 & 2 \\ 3 & 0 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -1 & 2 \\ 3 & 0 \end{array}\right] \) is \( \lambda^{2} + \lambda - 6 \).

The eigenvalues of \( \left[\begin{array}{cc} -1 & 2 \\ 3 & 0 \end{array}\right] \) are \( -3 \) and \( 2 \).


Example 17 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 6 & 2 \\ -12 & -8 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 6 & 2 \\ -12 & -8 \end{array}\right] \) is \( \lambda^{2} + 2 \lambda - 24 \).

The eigenvalues of \( \left[\begin{array}{cc} 6 & 2 \\ -12 & -8 \end{array}\right] \) are \( 4 \) and \( -6 \).


Example 18 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 3 & 2 \\ 9 & 0 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 3 & 2 \\ 9 & 0 \end{array}\right] \) is \( \lambda^{2} - 3 \lambda - 18 \).

The eigenvalues of \( \left[\begin{array}{cc} 3 & 2 \\ 9 & 0 \end{array}\right] \) are \( -3 \) and \( 6 \).


Example 19 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 0 & 1 \\ 12 & 1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 0 & 1 \\ 12 & 1 \end{array}\right] \) is \( \lambda^{2} - \lambda - 12 \).

The eigenvalues of \( \left[\begin{array}{cc} 0 & 1 \\ 12 & 1 \end{array}\right] \) are \( -3 \) and \( 4 \).


Example 20 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 2 & 2 \\ 4 & 0 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 2 & 2 \\ 4 & 0 \end{array}\right] \) is \( \lambda^{2} - 2 \lambda - 8 \).

The eigenvalues of \( \left[\begin{array}{cc} 2 & 2 \\ 4 & 0 \end{array}\right] \) are \( -2 \) and \( 4 \).


Example 21 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 7 & 2 \\ -6 & 0 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 7 & 2 \\ -6 & 0 \end{array}\right] \) is \( \lambda^{2} - 7 \lambda + 12 \).

The eigenvalues of \( \left[\begin{array}{cc} 7 & 2 \\ -6 & 0 \end{array}\right] \) are \( 3 \) and \( 4 \).


Example 22 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array}\right] \) is \( \lambda^{2} + \lambda - 2 \).

The eigenvalues of \( \left[\begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array}\right] \) are \( -2 \) and \( 1 \).


Example 23 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 9 & 2 \\ -42 & -11 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 9 & 2 \\ -42 & -11 \end{array}\right] \) is \( \lambda^{2} + 2 \lambda - 15 \).

The eigenvalues of \( \left[\begin{array}{cc} 9 & 2 \\ -42 & -11 \end{array}\right] \) are \( 3 \) and \( -5 \).


Example 24 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 4 & 2 \\ 0 & 2 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 4 & 2 \\ 0 & 2 \end{array}\right] \) is \( \lambda^{2} - 6 \lambda + 8 \).

The eigenvalues of \( \left[\begin{array}{cc} 4 & 2 \\ 0 & 2 \end{array}\right] \) are \( 2 \) and \( 4 \).


Example 25 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right] \) is \( \lambda^{2} + 9 \lambda + 18 \).

The eigenvalues of \( \left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right] \) are \( -6 \) and \( -3 \).


Example 26 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -3 & 1 \\ 12 & 1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -3 & 1 \\ 12 & 1 \end{array}\right] \) is \( \lambda^{2} + 2 \lambda - 15 \).

The eigenvalues of \( \left[\begin{array}{cc} -3 & 1 \\ 12 & 1 \end{array}\right] \) are \( -5 \) and \( 3 \).


Example 27 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 1 & 1 \\ 9 & 1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 1 & 1 \\ 9 & 1 \end{array}\right] \) is \( \lambda^{2} - 2 \lambda - 8 \).

The eigenvalues of \( \left[\begin{array}{cc} 1 & 1 \\ 9 & 1 \end{array}\right] \) are \( -2 \) and \( 4 \).


Example 28 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -3 & 1 \\ -3 & -7 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -3 & 1 \\ -3 & -7 \end{array}\right] \) is \( \lambda^{2} + 10 \lambda + 24 \).

The eigenvalues of \( \left[\begin{array}{cc} -3 & 1 \\ -3 & -7 \end{array}\right] \) are \( -4 \) and \( -6 \).


Example 29 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 9 & 1 \\ -42 & -8 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 9 & 1 \\ -42 & -8 \end{array}\right] \) is \( \lambda^{2} - \lambda - 30 \).

The eigenvalues of \( \left[\begin{array}{cc} 9 & 1 \\ -42 & -8 \end{array}\right] \) are \( 6 \) and \( -5 \).


Example 30 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right] \) is \( \lambda^{2} + 9 \lambda + 18 \).

The eigenvalues of \( \left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right] \) are \( -6 \) and \( -3 \).


Example 31 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 0 & 2 \\ -4 & -6 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 0 & 2 \\ -4 & -6 \end{array}\right] \) is \( \lambda^{2} + 6 \lambda + 8 \).

The eigenvalues of \( \left[\begin{array}{cc} 0 & 2 \\ -4 & -6 \end{array}\right] \) are \( -4 \) and \( -2 \).


Example 32 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right] \) is \( \lambda^{2} + 5 \lambda + 6 \).

The eigenvalues of \( \left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right] \) are \( -3 \) and \( -2 \).


Example 33 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 8 & 1 \\ -6 & 3 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 8 & 1 \\ -6 & 3 \end{array}\right] \) is \( \lambda^{2} - 11 \lambda + 30 \).

The eigenvalues of \( \left[\begin{array}{cc} 8 & 1 \\ -6 & 3 \end{array}\right] \) are \( 5 \) and \( 6 \).


Example 34 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 5 & 1 \\ -7 & -3 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 5 & 1 \\ -7 & -3 \end{array}\right] \) is \( \lambda^{2} - 2 \lambda - 8 \).

The eigenvalues of \( \left[\begin{array}{cc} 5 & 1 \\ -7 & -3 \end{array}\right] \) are \( 4 \) and \( -2 \).


Example 35 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 6 & 1 \\ -6 & 1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 6 & 1 \\ -6 & 1 \end{array}\right] \) is \( \lambda^{2} - 7 \lambda + 12 \).

The eigenvalues of \( \left[\begin{array}{cc} 6 & 1 \\ -6 & 1 \end{array}\right] \) are \( 4 \) and \( 3 \).


Example 36 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 6 & 1 \\ -11 & -6 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 6 & 1 \\ -11 & -6 \end{array}\right] \) is \( \lambda^{2} - 25 \).

The eigenvalues of \( \left[\begin{array}{cc} 6 & 1 \\ -11 & -6 \end{array}\right] \) are \( 5 \) and \( -5 \).


Example 37 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 7 & 2 \\ -5 & 0 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 7 & 2 \\ -5 & 0 \end{array}\right] \) is \( \lambda^{2} - 7 \lambda + 10 \).

The eigenvalues of \( \left[\begin{array}{cc} 7 & 2 \\ -5 & 0 \end{array}\right] \) are \( 5 \) and \( 2 \).


Example 38 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 3 & 1 \\ 3 & 5 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 3 & 1 \\ 3 & 5 \end{array}\right] \) is \( \lambda^{2} - 8 \lambda + 12 \).

The eigenvalues of \( \left[\begin{array}{cc} 3 & 1 \\ 3 & 5 \end{array}\right] \) are \( 2 \) and \( 6 \).


Example 39 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 1 & 2 \\ 8 & 1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 1 & 2 \\ 8 & 1 \end{array}\right] \) is \( \lambda^{2} - 2 \lambda - 15 \).

The eigenvalues of \( \left[\begin{array}{cc} 1 & 2 \\ 8 & 1 \end{array}\right] \) are \( -3 \) and \( 5 \).


Example 40 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 0 & 1 \\ -15 & -8 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 0 & 1 \\ -15 & -8 \end{array}\right] \) is \( \lambda^{2} + 8 \lambda + 15 \).

The eigenvalues of \( \left[\begin{array}{cc} 0 & 1 \\ -15 & -8 \end{array}\right] \) are \( -3 \) and \( -5 \).


Example 41 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right] \) is \( \lambda^{2} + 5 \lambda + 6 \).

The eigenvalues of \( \left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right] \) are \( -3 \) and \( -2 \).


Example 42 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -4 & 1 \\ 14 & 1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -4 & 1 \\ 14 & 1 \end{array}\right] \) is \( \lambda^{2} + 3 \lambda - 18 \).

The eigenvalues of \( \left[\begin{array}{cc} -4 & 1 \\ 14 & 1 \end{array}\right] \) are \( -6 \) and \( 3 \).


Example 43 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 9 & 2 \\ -15 & -2 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 9 & 2 \\ -15 & -2 \end{array}\right] \) is \( \lambda^{2} - 7 \lambda + 12 \).

The eigenvalues of \( \left[\begin{array}{cc} 9 & 2 \\ -15 & -2 \end{array}\right] \) are \( 3 \) and \( 4 \).


Example 44 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 5 & 1 \\ -21 & -5 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 5 & 1 \\ -21 & -5 \end{array}\right] \) is \( \lambda^{2} - 4 \).

The eigenvalues of \( \left[\begin{array}{cc} 5 & 1 \\ -21 & -5 \end{array}\right] \) are \( 2 \) and \( -2 \).


Example 45 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -1 & 2 \\ 6 & -2 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -1 & 2 \\ 6 & -2 \end{array}\right] \) is \( \lambda^{2} + 3 \lambda - 10 \).

The eigenvalues of \( \left[\begin{array}{cc} -1 & 2 \\ 6 & -2 \end{array}\right] \) are \( -5 \) and \( 2 \).


Example 46 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 0 & 1 \\ 10 & 3 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 0 & 1 \\ 10 & 3 \end{array}\right] \) is \( \lambda^{2} - 3 \lambda - 10 \).

The eigenvalues of \( \left[\begin{array}{cc} 0 & 1 \\ 10 & 3 \end{array}\right] \) are \( -2 \) and \( 5 \).


Example 47 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} -3 & 2 \\ 7 & 2 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} -3 & 2 \\ 7 & 2 \end{array}\right] \) is \( \lambda^{2} + \lambda - 20 \).

The eigenvalues of \( \left[\begin{array}{cc} -3 & 2 \\ 7 & 2 \end{array}\right] \) are \( -5 \) and \( 4 \).


Example 48 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 3 & 2 \\ 6 & -1 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 3 & 2 \\ 6 & -1 \end{array}\right] \) is \( \lambda^{2} - 2 \lambda - 15 \).

The eigenvalues of \( \left[\begin{array}{cc} 3 & 2 \\ 6 & -1 \end{array}\right] \) are \( -3 \) and \( 5 \).


Example 49 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 9 & 2 \\ -36 & -9 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 9 & 2 \\ -36 & -9 \end{array}\right] \) is \( \lambda^{2} - 9 \).

The eigenvalues of \( \left[\begin{array}{cc} 9 & 2 \\ -36 & -9 \end{array}\right] \) are \( 3 \) and \( -3 \).


Example 50 πŸ”—

Explain how to find the eigenvalues of the matrix \( \left[\begin{array}{cc} 4 & 1 \\ 6 & 3 \end{array}\right] \).

Answer:

The characteristic polynomial of \( \left[\begin{array}{cc} 4 & 1 \\ 6 & 3 \end{array}\right] \) is \( \lambda^{2} - 7 \lambda + 6 \).

The eigenvalues of \( \left[\begin{array}{cc} 4 & 1 \\ 6 & 3 \end{array}\right] \) are \( 1 \) and \( 6 \).