91Ó°ÊÓ

Let $$\begin{array}{l}A=\left[\begin{array}{rrr}1 & 2 & -1 \\\1 & 1 & 1 \\\1 & -1 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr}1 & -1 & 0 \\\1 & 1 & 1 \\\1 & 2 & -1 \end{array}\right] \\\C=\left[\begin{array}{rrr}1 & 2 & -1 \\\1 & 1 & 1 \\\2 & 1 & -1 \end{array}\right], \quad D=\left[\begin{array}{rrr}1 & 2 & -1 \\\\-3 & -1 & 3 \\\2 & 1 & -1\end{array}\right]\end{array}$$ In each case, find an elementary matrix E that satisfies the given equation. $$E A=C$$

Let \(\ell\) be a line through the origin in \(\mathbb{R}^{2}, P_{\ell}\) the linear transformation that projects a vector onto \(\ell\), and \(F_{\ell}\) the transformation that reflects a vector in \(\ell\) (a) Draw diagrams to show that \(F_{\ell}\) is linear. (b) Figure 3.14 suggests a way to find the matrix of \(F_{c}\) using the fact that the diagonals of a parallelogram bisect each other. Prove that \(F_{\ell}(\mathbf{x})=2 P_{\ell}(\mathbf{x})-\mathbf{x}\) and use this result to show that the standard matrix of \(F_{\ell}\) is \\[ \frac{1}{d_{1}^{2}+d_{2}^{2}}\left[\begin{array}{cc} d_{1}^{2}-d_{2}^{2} & 2 d_{1} d_{2} \\ 2 d_{1} d_{2} & -d_{1}^{2}+d_{2}^{2} \end{array}\right] \\] (where the direction vector of \(\ell\) is \(\mathrm{d}=\left[\begin{array}{l}d_{1} \\ d_{2}\end{array}\right]\) ). (c) If the angle between \(\ell\) and the positive \(x\) -axis is \(\theta\) show that the matrix of \(F_{\ell}\) is \\[ \left[\begin{array}{rr} \cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & -\cos 2 \theta \end{array}\right] \\]

Give a componentwise definition of a skew-symmetric matrix.

A square matrix \(A\) is called idempotent if \(A^{2}=A\). (The word idemporent comes from the Latin idem, meaning "same," and potere, meaning "to have power. Thus, something that is idempotent has the "same power" when squared. (a) Find three idempotent \(2 \times 2\) matrices. (b) Prove that the only invertible idempotent \(n \times n\) matrix is the identity matrix.