91Ó°ÊÓ

Find all matrices that commute with the given matrix \(A\). $$A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]$$

Find all matrices that commute with the given matrix \(A\). $$A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]$$

Find all matrices that commute with the given matrix \(A\). $$A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{array}\right]$$

a. Find the scaling matrix \(A\) that transforms \(\left[\begin{array}{r}2 \\\ -1\end{array}\right]\) into \(\left[\begin{array}{r}8 \\ -4\end{array}\right]\) b. Find the orthogonal projection matrix \(B\) that transforms \(\left[\begin{array}{l}2 \\ 3\end{array}\right]\) into \(\left[\begin{array}{l}2 \\ 0\end{array}\right]\) c. Find the rotation matrix \(C\) that transforms \(\left[\begin{array}{l}0 \\\ 5\end{array}\right]\) into \(\left[\begin{array}{l}3 \\ 4\end{array}\right]\) d. Find the shear matrix \(D\) that transforms \(\left[\begin{array}{l}1 \\\ 3\end{array}\right]\) into \(\left[\begin{array}{l}7 \\ 3\end{array}\right]\) e. Find the reflection matrix \(E\) that transforms \(\left[\begin{array}{l}7 \\\ 1\end{array}\right]\) into \(\left[\begin{array}{r}-5 \\ 5\end{array}\right]\)

Prove the distributive laws for matrices: $$A(C+D)=A C+A D$$ and $$(A+B) C=A C+B C$$.

Consider the rotation matrix \(D=\left[\begin{array}{cr}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]\) and the vector \(\vec{v}=\left[\begin{array}{l}\cos \beta \\ \sin \beta\end{array}\right],\) where \(\alpha\) and \(\beta\) are arbitrary angles. a. Draw a sketch to explain why \(D \vec{v}=\left[\begin{array}{c}\cos (\alpha+\beta) \\ \sin (\alpha+\beta)\end{array}\right]\) b. Compute \(D \vec{v}\). Use the result to derive the addition theorems for sine and cosine: \\[\cos (\alpha+\beta)=\ldots ., \quad \sin (\alpha+\beta)=\ldots\\]

Find all matrices \(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) such that \(a d-b c=1\) and \(A^{-1}=A\).

Consider the diagonal matrix \\[A=\left[\begin{array}{lll} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right].\\] a. For which values of \(a, b,\) and \(c\) is \(A\) invertible? If it is invertible, what is \(A^{-1} ?\) b. For which values of the diagonal elements is a diagonal matrix (of arbitrary size) invertible?

a. Consider the upper triangular \(3 \times 3\) matrix \\[A=\left[\begin{array}{lll} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{array}\right].\\] For which values of \(a, b, c, d, e,\) and \(f\) is \(A\) invert ible? b. More generally, when is an upper triangular matrix (of arbitrary size) invertible? c. If an upper triangular matrix is invertible, is its inverse an upper triangular matrix as well? d. When is a lower triangular matrix invertible?

If \(A\) is an invertible matrix and \(c\) is a nonzero scalar, is the matrix \(c A\) invertible? If so, what is the relationship between \(A^{-1}\) and \((c A)^{-1} ?\)