91Ó°ÊÓ

Find a \(P^{T} L U\) factorization of the given matrix \(A\). $$A=\left[\begin{array}{rrr} 0 & 1 & 4 \\ -1 & 2 & 1 \\ 1 & 3 & 3 \end{array}\right]$$

let \\[A=\left[\begin{array}{rrr}1 & 0 & -2 \\\\-3 & 1 & 1 \\\2 & 0 & -1\end{array}\right]\\] and \\[B=\left[\begin{array}{rrr}2 & 3 & 0 \\\1 & -1 & 1 \\\\-1 & 6 & 4\end{array}\right]\\]. Use the matrix-column representation of the product to write each column of \(A B\) as a linear combination of the columns of \(A\).

Find a \(P^{T} L U\) factorization of the given matrix \(A\). $$A=\left[\begin{array}{rrrr} 0 & 0 & 1 & 2 \\ -1 & 1 & 3 & 2 \\ 0 & 2 & 1 & 1 \\ 1 & 1 & -1 & 0 \end{array}\right]$$

If \(B=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) find conditions on \(a, b\) \(c,\) and \(d\) such that \(A B=B A\). $$A=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right]$$

let \\[A=\left[\begin{array}{rrr}1 & 0 & -2 \\\\-3 & 1 & 1 \\\2 & 0 & -1\end{array}\right]\\] and \\[B=\left[\begin{array}{rrr}2 & 3 & 0 \\\1 & -1 & 1 \\\\-1 & 6 & 4\end{array}\right]\\]. Use the row-matrix representation of the product to write each row of \(A B\) as a linear combination of the rows of \(B\).

Find the standard matrix of the given linear transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\). Reflection in the line \(y=x\)

Find the standard matrix of the given linear transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\). Reflection in the line \(y=-x\)

If \(B=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) find conditions on \(a, b\) \(c,\) and \(d\) such that \(A B=B A\). $$A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]$$

let \\[A=\left[\begin{array}{rrr}1 & 0 & -2 \\\\-3 & 1 & 1 \\\2 & 0 & -1\end{array}\right]\\] and \\[B=\left[\begin{array}{rrr}2 & 3 & 0 \\\1 & -1 & 1 \\\\-1 & 6 & 4\end{array}\right]\\]. Compute the outer product expansion of \(A B\).

Find a \(P^{T} L U\) factorization of the given matrix \(A\). $$A=\left[\begin{array}{rrrr} 0 & -1 & 1 & 3 \\ -1 & 1 & 1 & 2 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 1 \end{array}\right]$$