1. Problem: Show regularity and find inverse of the transformation with $$A=\begin{bmatrix}2&1&1\\1&1&-1\\1&2&0\end{bmatrix}$$ and $Y=AX$. 2. Compute determinant of $A$. 3. $$\det(A)=2(1\cdot0-2\cdot(-1))-1(1\cdot0-2\cdot1)+1(1\cdot(-1)-1\cdot1)=4$$. 4. Since $$\det(A)=4\neq 0$$ the transformation is regular. 5. Compute cofactors and adjugate. 6. Cofactor matrix is $$C=\begin{bmatrix}2&-1&1\\2&-1&-3\\-2&3&1\end{bmatrix}$$ and $$\operatorname{adj}(A)=C^{T}=\begin{bmatrix}2&2&-2\\-1&-1&3\\1&-3&1\end{bmatrix}$$. 7. Therefore $$A^{-1}=\frac{1}{4}\begin{bmatrix}2&2&-2\\-1&-1&3\\1&-3&1\end{bmatrix}$$ and the inverse transformation is $X=A^{-1}Y$. 1. Problem: Find the inverse transformation for $$A=\begin{bmatrix}1&2&5\\2&4&11\\-1&0&1\end{bmatrix}$$ where $Y=AX$. 2. Compute determinant of $A$. 3. $$\det(A)=1\cdot\det\begin{bmatrix}4&11\\0&1\end{bmatrix}-2\cdot\det\begin{bmatrix}2&11\\-1&1\end{bmatrix}+5\cdot\det\begin{bmatrix}2&4\\-1&0\end{bmatrix}=4-26+20=-2$$. 4. Since $$\det(A)=-2\neq0$$ the inverse exists. 5. Compute cofactor matrix and adjugate. 6. Cofactor matrix is $$C=\begin{bmatrix}4&-13&4\\-2&6&-2\\2&-1&0\end{bmatrix}$$ and $$\operatorname{adj}(A)=C^{T}=\begin{bmatrix}4&-2&2\\-13&6&-1\\4&-2&0\end{bmatrix}$$. 7. Thus $$A^{-1}=-\tfrac{1}{2}\begin{bmatrix}4&-2&2\\-13&6&-1\\4&-2&0\end{bmatrix}=\begin{bmatrix}-2&1&-1\\13/2&-3&1/2\\-2&1&0\end{bmatrix}$$ and the inverse transformation is $X=A^{-1}Y$. 1. Problem: For the rotation matrix $$A=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix}$$ show $A^{T}A=I$ and give the inverse. 2. Compute $A^{T}A$ explicitly. 3. $$A^{T}A=\begin{bmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{bmatrix}\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix}=\begin{bmatrix}\cos^{2}\alpha+\sin^{2}\alpha&0\\0&\sin^{2}\alpha+\cos^{2}\alpha\end{bmatrix}=I$$. 4. Therefore $A$ is orthogonal and $$A^{-1}=A^{T}=\begin{bmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{bmatrix}$$. 1. Problem: Represent transformations and find composite for $$X=AY,\ Y=BZ$$ with $$A=\begin{bmatrix}3&2\\1&1\end{bmatrix},\ B=\begin{bmatrix}-1&4\\1&0\end{bmatrix}$$. 2. Compute composite matrix $AB$. 3. $$AB=\begin{bmatrix}3&2\\1&1\end{bmatrix}\begin{bmatrix}-1&4\\1&0\end{bmatrix}=\begin{bmatrix}3(-1)+2(1)&3(4)+2(0)\\1(-1)+1(1)&1(4)+1(0)\end{bmatrix}=\begin{bmatrix}-1&12\\0&4\end{bmatrix}$$. 4. Hence $$X=ABZ=\begin{bmatrix}-1&12\\0&4\end{bmatrix}Z$$. 1. Problem: For $$A=\begin{bmatrix}1&0&2\\2&1&1\\1&2&0\end{bmatrix},\ B=\begin{bmatrix}1&1&1\\3&2&1\\5&3&1\end{bmatrix}$$ compute composite $BA$. 2. Multiply $B$ and $A$. 3. $$BA=\begin{bmatrix}1&1&1\\3&2&1\\5&3&1\end{bmatrix}\begin{bmatrix}1&0&2\\2&1&1\\1&2&0\end{bmatrix}=\begin{bmatrix}4&3&3\\8&4&8\\12&5&13\end{bmatrix}$$. 4. So the composite transformation is $Z=(BA)X$ with $BA$ as above. 1. Problem: Express $x_{1}+x_{2}+x_{3}$ in terms of $z_{1},z_{2}$ given the two transformations as written. 2. The notation given for the first transformation is ambiguous from the statement, so the explicit 2×3 matrix entries are not uniquely determined by the provided text. 3. I need the clear numeric entries of the first 2×3 matrix that maps $X\mapsto Y$ to proceed. 4. Please provide the exact 2×3 matrix in row-wise form so I can invert the mappings and express $x_{1}+x_{2}+x_{3}$ in terms of $z_{1},z_{2}$. 1. Problem: Show transformation is non-singular and find inverse for $$A=\begin{bmatrix}1&1&2\\2&0&1\\-1&1&1\end{bmatrix}$$ defined by $Y=AX$ as interpreted from the equations. 2. Compute determinant. 3. $$\det(A)=1\cdot\det\begin{bmatrix}0&1\\1&1\end{bmatrix}-1\cdot\det\begin{bmatrix}2&1\\-1&1\end{bmatrix}+2\cdot\det\begin{bmatrix}2&0\\-1&1\end{bmatrix}=-1-3+4=0$$. 4. Because $\det(A)=0$ the matrix is singular and no inverse exists for this interpreted matrix. 5. If the printed matrix differs from the intended one, please supply the exact matrix entries so I can recompute and, if nonzero determinant, produce $A^{-1}$. 1. Problem: Verify if the matrix $$A=\begin{bmatrix}3&4&2\\1&3&3\\9&1&1\end{bmatrix}$$ is orthogonal. 2. A quick necessary condition for orthogonality is that each column has norm 1 and columns are mutually orthogonal. 3. The squared norm of the first column is $3^{2}+1^{2}+9^{2}=91\neq1$ so the matrix is not orthogonal. 4. Therefore $A^{T}A\neq I$ and $A$ is not orthogonal. 1. Problem: Verify if the matrix $$A=\begin{bmatrix}\sin\theta&0&\cos\theta\\0&1&0\\\cos\theta&0&-\sin\theta\end{bmatrix}$$ is orthogonal. 2. Check column orthonormality. 3. The columns are $[\sin\theta,0,\cos\theta]^{T}$, $[0,1,0]^{T}$, $[\cos\theta,0,-\sin\theta]^{T}$ and they satisfy orthogonality and unit norms because $\sin^{2}\theta+\cos^{2}\theta=1$ and pairwise dot products vanish. 4. Hence $A^{T}A=I$ and $A$ is orthogonal. 1. Problem: Prove orthogonality of the integer matrix given in the statement. 2. The matrix as typed in the problem statement is ambiguous in formatting, so I cannot safely parse its entries to compute $A^{T}A$. 3. Please provide that 3×3 matrix in clear row-wise numeric form so I can compute $A^{T}A$ explicitly and confirm orthogonality or otherwise.