91影视

Question:If A is an invertible matrix andcis a nonzero scalar, is the matrixcA invertible? If so, what is the relationship betweenA^-1 and( cA )^-1?

The determinant of amatrix$\mathbf{\left[}\begin{array}{cc}\mathbf{a}& \mathbf{b}\\ \mathbf{c}& \mathbf{d}\end{array}\mathbf{\right]}\mathbf{\text{鈥�}}$is$\mathit{a}\mathit{d}\mathbf{鈭�}\mathit{b}\mathit{c}$ (we have seen this quantity inExercise $\mathbf{2}\mathbf{.1.13}$alreadyalready).Find the determinant of a matrix thatrepresents.$\mathit{a}\mathbf{\left(}\mathbf{n}\mathbf{\right)}$

a. orthogonal projection

b. reflection about a line

c. rotation

d. (horizontal or vertical) shear.

What do your answers tell you about the invertibility of these matrices?

If A and B are two 4 脳 3 matrices such that$\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}\mathbf{B}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$ for all vectors$\stackrel{\mathbf{鈫�}}{\mathbf{v}}$ in${\mathbf{鈩�}}^{\mathbf{3}}$ , then matrices A and B must be equal.

Question:Find A^-1 for $\mathit{A}\mathbf{=}\left(\begin{array}{cc}1& k\\ 0& 鈭�1\end{array}\right)$.

If matrices A and B commute, then the formula A²B = BA² must hold.

Describe each of the linear transformations defined by the matrices in parts (a) through (c) geometrically, as a well-known transformation combined with a scaling. Give the scaling factor in each case.

$\mathit{a}\mathbf{.}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{1}\end{array}\mathbf{\right]}\mathbf{\text{鈥夆赌夆赌夆赌夆赌夆赌夆赌�}}\mathit{b}\mathbf{.}\mathbf{\left[}\begin{array}{cc}\mathbf{3}& \mathbf{0}\\ \mathbf{鈭�}\mathbf{1}& \mathbf{3}\end{array}\mathbf{\right]}\mathbf{\text{鈥夆赌夆赌夆赌夆赌夆赌夆赌�}}\mathit{c}\mathbf{.}\mathbf{\left[}\begin{array}{cc}\mathbf{3}& \mathbf{4}\\ \mathbf{4}& \mathbf{鈭�}\mathbf{3}\end{array}\mathbf{\right]}$

Question:Consider a square matrix that differs from the identity matrix at just one entry, off the diagonal, for example,

$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 鈭�\frac{1}{2}& 0& 1\end{array}\right)$

In general, is a matrix M of this form invertible? If so, what is the M^-1?

For the matrix A in exercise 33 through 42 , Compute $A^2=AA$, and localid="1664204847004" $A^3=AAA$and localid="1664205252603" $A^4$. Describe the pattern that emerges, and use this pattern to find $A^{1001}$.Interpret your answers geometrically, in terms of rotations, reflections, shears, and orthogonal projections.

$A=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right]$

The function $\mathit{T}\left[\begin{array}{c}x\\ y\end{array}\right]\mathbf{=}\left[\begin{array}{c}x-y\\ y-x\end{array}\right]$is a linear transformation?

If possible, compute the matrix products in Exercises 1 through 13, using paper and pencil.

3. $\mathbf{\left[}\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\mathbf{\right]}\mathbf{\left[}\begin{array}{cc}1& 2\\ 3& 4\end{array}\mathbf{\right]}$