1. **Vector Operations:** Given vectors \(\mathbf{u} = \begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix}4 \\ 5 \\ 6\end{pmatrix}\). 2. **Vector sum \(\mathbf{u} + \mathbf{v}\):** \[ \mathbf{u} + \mathbf{v} = \begin{pmatrix}1 + 4 \\ 2 + 5 \\ 3 + 6\end{pmatrix} = \begin{pmatrix}5 \\ 7 \\ 9\end{pmatrix} \] 3. **Dot product \(\mathbf{u} \cdot \mathbf{v}\):** \[ \mathbf{u} \cdot \mathbf{v} = 1 \times 4 + 2 \times 5 + 3 \times 6 = 4 + 10 + 18 = 32 \] The dot product measures how much two vectors are aligned; it is a scalar. 4. **Cross product \(\mathbf{u} \times \mathbf{v}\):** \[ \mathbf{u} \times \mathbf{v} = \begin{pmatrix} 2 \times 6 - 3 \times 5 \\ 3 \times 4 - 1 \times 6 \\ 1 \times 5 - 2 \times 4 \end{pmatrix} = \begin{pmatrix} 12 - 15 \\ 12 - 6 \\ 5 - 8 \end{pmatrix} = \begin{pmatrix} -3 \\ 6 \\ -3 \end{pmatrix} \] The cross product results in a vector perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\). 5. **Matrix Multiplication:** Given matrices \(A = \begin{pmatrix}1 & 3 \\ 2 & 4\end{pmatrix}\) and \(B = \begin{pmatrix}5 & 7 \\ 6 & 8\end{pmatrix}\). 6. **Calculate \(AB\):** \[ AB = \begin{pmatrix}1 \times 5 + 3 \times 6 & 1 \times 7 + 3 \times 8 \\ 2 \times 5 + 4 \times 6 & 2 \times 7 + 4 \times 8\end{pmatrix} = \begin{pmatrix}5 + 18 & 7 + 24 \\ 10 + 24 & 14 + 32\end{pmatrix} = \begin{pmatrix}23 & 31 \\ 34 & 46\end{pmatrix} \] 7. **Calculate \(BA\):** \[ BA = \begin{pmatrix}5 \times 1 + 7 \times 2 & 5 \times 3 + 7 \times 4 \\ 6 \times 1 + 8 \times 2 & 6 \times 3 + 8 \times 4\end{pmatrix} = \begin{pmatrix}5 + 14 & 15 + 28 \\ 6 + 16 & 18 + 32\end{pmatrix} = \begin{pmatrix}19 & 43 \\ 22 & 50\end{pmatrix} \] 8. **Compare \(AB\) and \(BA\):** They are not equal, so matrix multiplication is not commutative. 9. **Determinants and Inverses:** Given matrix \(C = \begin{pmatrix}2 & 1 \\ 3 & 4\end{pmatrix}\). 10. **Calculate determinant of \(C\):** \[ \det(C) = 2 \times 4 - 1 \times 3 = 8 - 3 = 5 \] 11. **Invertibility of \(C\):** Because \(\det(C) \neq 0\), matrix \(C\) is invertible. 12. **Find inverse \(C^{-1}\):** \[ C^{-1} = \frac{1}{\det(C)} \begin{pmatrix}4 & -1 \\ -3 & 2\end{pmatrix} = \frac{1}{5} \begin{pmatrix}4 & -1 \\ -3 & 2\end{pmatrix} = \begin{pmatrix}0.8 & -0.2 \\ -0.6 & 0.4\end{pmatrix} \] 13. **Significance of determinant:** The determinant indicates whether a matrix is invertible. If \(\det \neq 0\), the matrix has an inverse; if zero, no inverse exists. The determinant also reflects the scaling factor of area or volume under the linear transformation defined by the matrix. **Final answers:** - \(\mathbf{u} + \mathbf{v} = \begin{pmatrix}5 \\ 7 \\ 9\end{pmatrix}\) - \(\mathbf{u} \cdot \mathbf{v} = 32\) - \(\mathbf{u} \times \mathbf{v} = \begin{pmatrix}-3 \\ 6 \\ -3\end{pmatrix}\) - \(AB = \begin{pmatrix}23 & 31 \\ 34 & 46\end{pmatrix}, BA = \begin{pmatrix}19 & 43 \\ 22 & 50\end{pmatrix}\) (not equal) - \(\det(C)=5\), invertible - \(C^{-1} = \begin{pmatrix}0.8 & -0.2 \\ -0.6 & 0.4\end{pmatrix}\)