Question

Problem 1. Let A = \begin{pmatrix} 1 & 2\\ 3 & 4 \end{pmatrix} which has an one eigenvalue equal to 5.74. Determine the other eigenvalue of this matrix. Problem 2. Let A = \begin{pmatrix} a & b\\ 0 & c \end{pmatrix}. Show that a is an eigenvalue with the eigenvector x = \begin{pmatrix} 1\\ 0 \end{pmatrix} of matrix A. Problem 3. Let A = \begin{pmatrix} 5 & -4\\ -4 & 5 \end{pmatrix}. By using the power formula for matrices A" = PD"P?¹ where P is the eigenvector matrix find ?A. Linear Algebra II (MTC37W1) 2024 Problem 4. Let A = \begin{pmatrix} 4 & 5\\ -3 & -4 \end{pmatrix}. Compute A^{1000001} Problem 5. Consider the matrix A = \begin{pmatrix} 4 & -3\\ 1 & 0 \end{pmatrix} (ii) Find a matrix P such that P?¹AP is diagonal. (iii) Find the eigenvalues and the determinant of A^{2008}.

          Problem 1.
Let A = \begin{pmatrix} 1 & 2\\ 3 & 4 \end{pmatrix} which has an one eigenvalue equal to 5.74. Determine the other eigenvalue of this matrix.
Problem 2.
Let A = \begin{pmatrix} a & b\\ 0 & c \end{pmatrix}. Show that a is an eigenvalue with the eigenvector x = \begin{pmatrix} 1\\ 0 \end{pmatrix} of matrix A.
Problem 3.
Let A = \begin{pmatrix} 5 & -4\\ -4 & 5 \end{pmatrix}. By using the power formula for matrices A" = PD"P?¹ where P is the eigenvector matrix find ?A.
Linear Algebra II (MTC37W1)  2024
Problem 4.
Let A = \begin{pmatrix} 4 & 5\\ -3 & -4 \end{pmatrix}. Compute A^{1000001}
Problem 5.
Consider the matrix A = \begin{pmatrix} 4 & -3\\ 1 & 0 \end{pmatrix}
(ii) Find a matrix P such that P?¹AP is diagonal.
(iii) Find the eigenvalues and the determinant of A^{2008}.

Problem 1.
Let A =
< p m a t r i x >
which has an one eigenvalue equal to 5.74. Determine the other eigenvalue of this matrix.
Problem 2.
Let A =
< p m a t r i x >
. Show that a is an eigenvalue with the eigenvector x =
< p m a t r i x >
of matrix A.
Problem 3.
Let A =
< p m a t r i x >
. By using the power formula for matrices A" = PD"P?¹ where P is the eigenvector matrix find ?A.
Linear Algebra II (MTC37W1) 2024
Problem 4.
Let A =
< p m a t r i x >
. Compute A^1000001
Problem 5.
Consider the matrix A =
< p m a t r i x >

(ii) Find a matrix P such that P?¹AP is diagonal.
(iii) Find the eigenvalues and the determinant of A^2008.

Added by Robert H.

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