Problem 36
A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{ccc} 2 & 0 & 3 \\ 1 & 1 & 1 \\ 3 & -1 & 2 \end{array}\right], \quad \vec{x}=\left[\begin{array}{l} 1 \\ 4 \\ 2 \end{array}\right] $$
Problem 37
A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{ccc} -2 & 0 & 3 \\ 1 & 1 & -2 \\ 4 & 2 & -1 \end{array}\right], \quad \vec{x}=\left[\begin{array}{l} 4 \\ 3 \\ 1 \end{array}\right] $$
Problem 38
A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{cc} 2 & -1 \\ 4 & 3 \end{array}\right], \quad \vec{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] $$
Problem 39
A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 1 & 0 & 2 \\ 2 & 3 & 1 \end{array}\right], \quad \vec{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] $$
Problem 40
Let \(A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] .\) Find \(A^{2}\) and \(A^{3}\).
Problem 41
Let \(A=\left[\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right] .\) Find \(A^{2}\) and \(A^{3}\).
Problem 42
Let \(A=\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5\end{array}\right] .\) Find \(A^{2}\) and \(A^{3} .\)
Problem 43
Let \(A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right] .\) Find \(A^{2}\) and \(A^{3}\).
Problem 44
Let \(A=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & 0\end{array}\right] .\) Find \(A^{2}\) and \(A^{3}\).
Problem 45
In the text we state that \((A+B)^{2} \neq\) \(A^{2}+2 A B+B^{2} .\) We investigate that claim here. (a) Let \(A=\left[\begin{array}{cc}5 & 3 \\ -3 & -2\end{array}\right]\) and let \(B=\) $$ \left[\begin{array}{cc} -5 & -5 \\ -2 & 1 \end{array}\right] . \text { Compute } A+B $$ (b) Find \((A+B)^{2}\) by using your answer from (a). (c) Compute \(A^{2}+2 A B+B^{2}\). (d) Are the results from (a) and (b) the same? (e) Carefully expand the expression \((A+B)^{2}=(A+B)(A+B)\) and show why this is not equal to \(A^{2}+2 A B+B^{2}\)
