+26400 Consider the matrices \(\begin{align*} \mathbf{A} &= \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \\ \mathbf{B} &= \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \\ \mathbf{C} &= \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix}. \end{align*}\)
Let \(v\) be a unit vector.
Find the magnitudes \(\|\mathbf{A}\mathbf{v}\|,\ \|\mathbf{B}\mathbf{v}\|,\ \|\mathbf{C}\mathbf{v}\|\),
or state if any of these cannot be uniquely determined from the information given.
\(\text{Let unit vector $v =\dbinom10$ }\)
\(\begin{array}{|rcll|} \hline \mathbf{A}\mathbf{v} &=& \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\cdot\dbinom10 \\ \mathbf{A}\mathbf{v} &=& \dbinom11 \\ \|\mathbf{A}\mathbf{v}\| &=& \sqrt{1^2+1^2} \\ \mathbf{\|\mathbf{A}\mathbf{v}\|} &=& \mathbf{\sqrt{2}} \\ \hline \end{array} \begin{array}{|rcll|} \hline \mathbf{B}\mathbf{v} &=& \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\cdot\dbinom10 \\ \mathbf{B}\mathbf{v} &=& \dbinom11 \\ \|\mathbf{B}\mathbf{v}\| &=& \sqrt{1^2+1^2} \\ \mathbf{\|\mathbf{B}\mathbf{v}\|} &=& \mathbf{\sqrt{2}} \\ \hline \end{array} \begin{array}{|rcll|} \hline \mathbf{C}\mathbf{v} &=& \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix}\cdot\dbinom10 \\ \mathbf{C}\mathbf{v} &=& \dbinom12 \\ \|\mathbf{C}\mathbf{v}\| &=& \sqrt{1^2+2^2} \\ \mathbf{\|\mathbf{C}\mathbf{v}\|} &=& \mathbf{\sqrt{5}} \\ \hline \end{array}\)
