91Ó°ÊÓ

Construct transformation matrices that represent the following rotations about the \(z\)-axis: (i) anticlockwise through \(45^{\circ}\), (ii) anticlockwise through \(90^{\circ}\), (iii) clockwise through \(90^{\circ}\).

Construct a transformation matrix that represents the interchange of \(x\) and \(y\) coordinates of a point.

For the following matrices, $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 3 \\ 0 & 3 & 4 \end{array}\right) \quad \mathbf{B}=\left(\begin{array}{rrr} 0 & 1 & -4 \\ 2 & -3 & 0 \end{array}\right) \quad \mathbf{C}=\left(\begin{array}{rr} -5 & 3 \\ 4 & -1 \\ 2 & -1 \end{array}\right) \quad \mathbf{D}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{array}\right) \\ &\mathbf{P}=\left(\begin{array}{rr} 1 & -2 \\ 0 & 4 \end{array}\right) \quad \mathbf{Q}=\left(\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right) \quad \mathbf{a}=\left(\begin{array}{r} 0 \\ -3 \\ 1 \end{array}\right) \mathbf{b}=\left(\begin{array}{lll} 2 & 5 & -2 \end{array}\right) \end{aligned} $$ find, if possible: \(\operatorname{det} \mathbf{A}, \operatorname{tr} \mathbf{A}\)

For the following matrices, $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 3 \\ 0 & 3 & 4 \end{array}\right) \quad \mathbf{B}=\left(\begin{array}{rrr} 0 & 1 & -4 \\ 2 & -3 & 0 \end{array}\right) \quad \mathbf{C}=\left(\begin{array}{rr} -5 & 3 \\ 4 & -1 \\ 2 & -1 \end{array}\right) \quad \mathbf{D}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{array}\right) \\ &\mathbf{P}=\left(\begin{array}{rr} 1 & -2 \\ 0 & 4 \end{array}\right) \quad \mathbf{Q}=\left(\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right) \quad \mathbf{a}=\left(\begin{array}{r} 0 \\ -3 \\ 1 \end{array}\right) \mathbf{b}=\left(\begin{array}{lll} 2 & 5 & -2 \end{array}\right) \end{aligned} $$ find, if possible: det \(\mathbf{D}\), tr \(\mathbf{D}\)

For the following matrices, $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 3 \\ 0 & 3 & 4 \end{array}\right) \quad \mathbf{B}=\left(\begin{array}{rrr} 0 & 1 & -4 \\ 2 & -3 & 0 \end{array}\right) \quad \mathbf{C}=\left(\begin{array}{rr} -5 & 3 \\ 4 & -1 \\ 2 & -1 \end{array}\right) \quad \mathbf{D}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{array}\right) \\ &\mathbf{P}=\left(\begin{array}{rr} 1 & -2 \\ 0 & 4 \end{array}\right) \quad \mathbf{Q}=\left(\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right) \quad \mathbf{a}=\left(\begin{array}{r} 0 \\ -3 \\ 1 \end{array}\right) \mathbf{b}=\left(\begin{array}{lll} 2 & 5 & -2 \end{array}\right) \end{aligned} $$ find, if possible: det \(\mathbf{P}\), tr \(\mathbf{P}\)

For the following matrices, $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 3 \\ 0 & 3 & 4 \end{array}\right) \quad \mathbf{B}=\left(\begin{array}{rrr} 0 & 1 & -4 \\ 2 & -3 & 0 \end{array}\right) \quad \mathbf{C}=\left(\begin{array}{rr} -5 & 3 \\ 4 & -1 \\ 2 & -1 \end{array}\right) \quad \mathbf{D}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{array}\right) \\ &\mathbf{P}=\left(\begin{array}{rr} 1 & -2 \\ 0 & 4 \end{array}\right) \quad \mathbf{Q}=\left(\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right) \quad \mathbf{a}=\left(\begin{array}{r} 0 \\ -3 \\ 1 \end{array}\right) \mathbf{b}=\left(\begin{array}{lll} 2 & 5 & -2 \end{array}\right) \end{aligned} $$ find, if possible: \(\mathbf{A}^{\top}\)

For the following matrices, $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 3 \\ 0 & 3 & 4 \end{array}\right) \quad \mathbf{B}=\left(\begin{array}{rrr} 0 & 1 & -4 \\ 2 & -3 & 0 \end{array}\right) \quad \mathbf{C}=\left(\begin{array}{rr} -5 & 3 \\ 4 & -1 \\ 2 & -1 \end{array}\right) \quad \mathbf{D}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{array}\right) \\ &\mathbf{P}=\left(\begin{array}{rr} 1 & -2 \\ 0 & 4 \end{array}\right) \quad \mathbf{Q}=\left(\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right) \quad \mathbf{a}=\left(\begin{array}{r} 0 \\ -3 \\ 1 \end{array}\right) \mathbf{b}=\left(\begin{array}{lll} 2 & 5 & -2 \end{array}\right) \end{aligned} $$ find, if possible: \(\mathbf{C}^{\top}\)

For the following matrices, $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 3 \\ 0 & 3 & 4 \end{array}\right) \quad \mathbf{B}=\left(\begin{array}{rrr} 0 & 1 & -4 \\ 2 & -3 & 0 \end{array}\right) \quad \mathbf{C}=\left(\begin{array}{rr} -5 & 3 \\ 4 & -1 \\ 2 & -1 \end{array}\right) \quad \mathbf{D}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{array}\right) \\ &\mathbf{P}=\left(\begin{array}{rr} 1 & -2 \\ 0 & 4 \end{array}\right) \quad \mathbf{Q}=\left(\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right) \quad \mathbf{a}=\left(\begin{array}{r} 0 \\ -3 \\ 1 \end{array}\right) \mathbf{b}=\left(\begin{array}{lll} 2 & 5 & -2 \end{array}\right) \end{aligned} $$ find, if possible: \(\mathbf{D}^{\top}\)

For the following matrices, $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 3 \\ 0 & 3 & 4 \end{array}\right) \quad \mathbf{B}=\left(\begin{array}{rrr} 0 & 1 & -4 \\ 2 & -3 & 0 \end{array}\right) \quad \mathbf{C}=\left(\begin{array}{rr} -5 & 3 \\ 4 & -1 \\ 2 & -1 \end{array}\right) \quad \mathbf{D}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{array}\right) \\ &\mathbf{P}=\left(\begin{array}{rr} 1 & -2 \\ 0 & 4 \end{array}\right) \quad \mathbf{Q}=\left(\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right) \quad \mathbf{a}=\left(\begin{array}{r} 0 \\ -3 \\ 1 \end{array}\right) \mathbf{b}=\left(\begin{array}{lll} 2 & 5 & -2 \end{array}\right) \end{aligned} $$ find, if possible: \(\mathbf{P}^{\top}\)

For the following matrices, $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 3 \\ 0 & 3 & 4 \end{array}\right) \quad \mathbf{B}=\left(\begin{array}{rrr} 0 & 1 & -4 \\ 2 & -3 & 0 \end{array}\right) \quad \mathbf{C}=\left(\begin{array}{rr} -5 & 3 \\ 4 & -1 \\ 2 & -1 \end{array}\right) \quad \mathbf{D}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{array}\right) \\ &\mathbf{P}=\left(\begin{array}{rr} 1 & -2 \\ 0 & 4 \end{array}\right) \quad \mathbf{Q}=\left(\begin{array}{ll} 3 & 0 \\ 0 & 1 \end{array}\right) \quad \mathbf{a}=\left(\begin{array}{r} 0 \\ -3 \\ 1 \end{array}\right) \mathbf{b}=\left(\begin{array}{lll} 2 & 5 & -2 \end{array}\right) \end{aligned} $$ find, if possible: \(\mathbf{a}^{\top}\)