Questão 2) Sejam as matrizes: $A=\left[\begin{array}{rrr}1 & 2 & 3 \\ 2 & -1 & 1\end{array}\right]$ $B=\left[\begin{array}{ccc}-2 & 0 & 1 \\ 3 & 0 & 1\end{array}\right]$ $C=\left[\begin{array}{c}-1 \\ 2 \\ 4\end{array}\right]$ $D=\left[\begin{array}{lll}2 & 0 & 1\end{array}\right]$ Calcule: a) $A^{t}+B^{t}$ c) A. C b) $2 A-3 B$ d) C.D

Step 1 ：Given matrices A, B, C, D as follows:

Step 2 ：\[A=\begin{bmatrix}1 & 2 & 3 \\ 2 & -1 & 1\end{bmatrix}\]

Step 3 ：\[B=\begin{bmatrix}-2 & 0 & 1 \\ 3 & 0 & 1\end{bmatrix}\]

Step 4 ：\[C=\begin{bmatrix}-1 \\ 2 \\ 4\end{bmatrix}\]

Step 5 ：\[D=\begin{bmatrix}2 & 0 & 1\end{bmatrix}\]

Step 6 ：For part a), we need to calculate the transpose of A and B and then add them. The transpose of a matrix is obtained by interchanging its rows into columns or columns into rows. So, we get:

Step 7 ：\[A^{t}=\begin{bmatrix}1 & 2 \\ 2 & -1 \\ 3 & 1\end{bmatrix}\]

Step 8 ：\[B^{t}=\begin{bmatrix}-2 & 3 \\ 0 & 0 \\ 1 & 1\end{bmatrix}\]

Step 9 ：Adding these two matrices, we get:

Step 10 ：\[A^{t}+B^{t}=\begin{bmatrix}-1 & 5 \\ 2 & -1 \\ 4 & 2\end{bmatrix}\]

Step 11 ：For part b), we need to multiply matrix A by 2 and matrix B by 3 and then subtract the two resulting matrices. We get:

Step 12 ：\[2A-3B=\begin{bmatrix}8 & 4 & 3 \\ -5 & -2 & -1\end{bmatrix}\]

Step 13 ：For part c), we need to multiply matrix A with matrix C. Matrix multiplication is done element by element and then adding them up. We get:

Step 14 ：\[A.C=\begin{bmatrix}15 \\ 0\end{bmatrix}\]

Step 15 ：For part d), we need to multiply matrix C with matrix D. We get:

Step 16 ：\[C.D=\begin{bmatrix}2\end{bmatrix}\]

Step 17 ：So, the final answers are:

Step 18 ：\[\boxed{A^{t}+B^{t}=\begin{bmatrix}-1 & 5 \\ 2 & -1 \\ 4 & 2\end{bmatrix}}\]

Step 19 ：\[\boxed{2A-3B=\begin{bmatrix}8 & 4 & 3 \\ -5 & -2 & -1\end{bmatrix}}\]

Step 20 ：\[\boxed{A.C=\begin{bmatrix}15 \\ 0\end{bmatrix}}\]

Step 21 ：\[\boxed{C.D=\begin{bmatrix}2\end{bmatrix}}\]