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-1 2 and y = 1 3 -y 6. x = -3. Compute (E) y. (-1)

-1 2 and y = 1 3 -y 6. x = -3. Compute (E) y. (-1)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do question 6
4. Let \( A = \begin{pmatrix} 3 & 1 \\ -3 & 1 \end{pmatrix} \). Find an invertible matrix \( P \) and a matrix \( C = \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \) so that \( A = PCP^{-1} \). 5. Let \( A = \begin{pmatrix} 7 & 11 & 20 & 17 \\ -20 & -40 & -86 & -74 \\ 0 & -5 & -10 & -10 \\ 0 & 28 & 60 & 53 \end{pmatrix} \). Find an invertible matrix \( P \) and a block diagonal matrix \( C = \begin{pmatrix} a & b & 0 & 0 \\ b & a & 0 & 0 \\ 0 & 0 & c & d \\ 0 & 0 & -d & c \end{pmatrix} \) so that \( A = PCP^{-1} \). 6. \( \mathbf{x} = \begin{pmatrix} -1 \\ 2 \end{pmatrix} \) and \( \mathbf{y} = \begin{pmatrix} 2 \\ -3 \end{pmatrix} \). Compute \(\frac{(\mathbf{x} \cdot \mathbf{y})}{\|\mathbf{y}\|^2}\mathbf{y}\). 7. Find a unit vector in the direction of \( \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} \). 8. Find a unit vector in the direction of \( \begin{pmatrix} -1 \\ 2 \\ -3 \end{pmatrix} \). 9. Find the distance from \( \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} \) to \( \begin{pmatrix} 2 \\ -3 \\ 2 \end{pmatrix} \).
Transcribed Image Text:4. Let \( A = \begin{pmatrix} 3 & 1 \\ -3 & 1 \end{pmatrix} \). Find an invertible matrix \( P \) and a matrix \( C = \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \) so that \( A = PCP^{-1} \). 5. Let \( A = \begin{pmatrix} 7 & 11 & 20 & 17 \\ -20 & -40 & -86 & -74 \\ 0 & -5 & -10 & -10 \\ 0 & 28 & 60 & 53 \end{pmatrix} \). Find an invertible matrix \( P \) and a block diagonal matrix \( C = \begin{pmatrix} a & b & 0 & 0 \\ b & a & 0 & 0 \\ 0 & 0 & c & d \\ 0 & 0 & -d & c \end{pmatrix} \) so that \( A = PCP^{-1} \). 6. \( \mathbf{x} = \begin{pmatrix} -1 \\ 2 \end{pmatrix} \) and \( \mathbf{y} = \begin{pmatrix} 2 \\ -3 \end{pmatrix} \). Compute \(\frac{(\mathbf{x} \cdot \mathbf{y})}{\|\mathbf{y}\|^2}\mathbf{y}\). 7. Find a unit vector in the direction of \( \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} \). 8. Find a unit vector in the direction of \( \begin{pmatrix} -1 \\ 2 \\ -3 \end{pmatrix} \). 9. Find the distance from \( \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} \) to \( \begin{pmatrix} 2 \\ -3 \\ 2 \end{pmatrix} \).
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