91影视

(a)

Consider the matrix,

$a\left(n\right)=\left[\begin{array}{cc}{a}^2& ab\\ ab& b^2\end{array}\right]$

The determinant of the matrix is,

$\begin{array}{c}\det \left[a\left(n\right)\right]=\det \left(\left[\begin{array}{cc}{a}^2& ab\\ ab& b^2\end{array}\right]\right)\\ 鈬�\det \left[a\left(n\right)\right]=a^2脳b^2鈭�ab脳ab\\ 鈬�\det \left[a\left(n\right)\right]=a^2{b}^2鈭�a^2{b}^2\\ 鈭�\det \left[a\left(n\right)\right]=0\end{array}$

As the determinant of the matrix is zero, thus, it is not invertible. The zero determinant of the matrix represents the orthogonal projection of .$0$

(b)

Consider the matrix,

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

The reflection matrix is,

$A_r=\left[\begin{array}{cc}a& b\\ b& 鈭�a\end{array}\right]$

The determinant of the matrix is,

$\begin{array}{c}\det \left[A_r\right]=\det \left(\left[\begin{array}{cc}a& b\\ b& 鈭�a\end{array}\right]\right)\\ 鈬�\det \left[A_r\right]=a脳鈭�a鈭�b脳b\\ 鈬�\det \left[A_r\right]=鈭�a^2鈭�b^2\\ 鈭�\det \left[A_r\right]=鈭�(a^2+b^2)\end{array}$

Thus,

$\begin{array}{l}\det \left[A_r\right]=鈭�(a^2+b^2)\\ 鈭�\det \left[A_r\right]=鈭�1\text{鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�}(鈭�a^2+b^2=1)\end{array}$

As the determinant of the matrix is nonzero, thus, it is invertible. The determinant of the matrix represents the reflection about a line of -1.

(c)

Consider the matrix,

$R=\left[\begin{array}{cc}\cos 胃& 鈭�\sin 胃\\ \sin 胃& \cos 胃\end{array}\right]$

The determinant of the matrix is,

$\begin{array}{c}\det \left[R\right]=\det \left(\left[\begin{array}{cc}\cos 胃& 鈭�\sin 胃\\ \sin 胃& \cos 胃\end{array}\right]\right)\\ 鈬�\det \left[R\right]=\cos 胃脳\cos 胃鈭�\sin 胃脳鈭�\sin 胃\\ 鈬�\det \left[R\right]={\cos }^2胃+{\sin }^2胃\\ 鈭�\det \left[R\right]=1\end{array}$

As the determinant of the matrix is nonzero, thus, it is invertible. The determinant of the matrix represents the rotation of $-1$.

(d)

Consider the matrix,

$K=\left[\begin{array}{cc}1& k\\ 0& 1\end{array}\right]or\left[\begin{array}{cc}1& 0\\ k& 1\end{array}\right]$

The determinant of the matrix is,

$\begin{array}{c}\det \left[K\right]=\det \left(\left[\begin{array}{cc}1& k\\ 0& 1\end{array}\right]\right)\\ 鈬�\det \left[K\right]=1脳1鈭�0脳k\\ 鈬�\det \left[K\right]=1鈭�0\\ 鈭�\det \left[K\right]=1\end{array}$

As the determinant of the matrix is nonzero, thus, it is invertible. The determinant of the matrix represents the horizontal/vertical of 1 .