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 \( 3 \).
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 \( -5 \).
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 \( -2 \).
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 \( 1 \).
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 \( -4 \).
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 \( 1 \).
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 \( -4 \).
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 \( 4 \).
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 \( -2 \).
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 \( 6 \).
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 \( 4 \).
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 \( 6 \).
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 \( 2 \).
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 \( 5 \).
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 \( 4 \).
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 \( 2 \).
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 \( -6 \).
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 \( 6 \).
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 \( 4 \).
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 \( 4 \).
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 \( 4 \).
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 \( 1 \).
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 \( -5 \).
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 \( 4 \).
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 \( -3 \).
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 \( 3 \).
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 \( 4 \).
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 \( -6 \).
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 \( -5 \).
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 \( -3 \).
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 \( -2 \).
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 \( -2 \).
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 \( 6 \).
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 \( -2 \).
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 \( 3 \).
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 \( -5 \).
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 \( 2 \).
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 \( 6 \).
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 \( 5 \).
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 \( -5 \).
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 \( -2 \).
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 \( 3 \).
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 \( 4 \).
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 \( -2 \).
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 \( 2 \).
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 \( 5 \).
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 \( 4 \).
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 \( 5 \).
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 \( -3 \).
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 \( 6 \).