1. **Problem 1(a): Find the rank of the matrix** $$\begin{bmatrix}1 & 2 & 1 & 2 \\ 1 & 3 & 2 & 4 \\ 2 & 4 & 3 & 4 \\ 3 & 7 & 5 & 6\end{bmatrix}$$ 2. **Formula and rules:** The rank of a matrix is the number of non-zero rows in its row echelon form (REF) or reduced row echelon form (RREF). 3. **Step-by-step row reduction:** - Start with the matrix: $$\begin{bmatrix}1 & 2 & 1 & 2 \\ 1 & 3 & 2 & 4 \\ 2 & 4 & 3 & 4 \\ 3 & 7 & 5 & 6\end{bmatrix}$$ - Subtract row 1 from row 2: $$R_2 = R_2 - R_1 \Rightarrow \begin{bmatrix}1 & 2 & 1 & 2 \\ 0 & 1 & 1 & 2 \\ 2 & 4 & 3 & 4 \\ 3 & 7 & 5 & 6\end{bmatrix}$$ - Subtract 2 times row 1 from row 3: $$R_3 = R_3 - 2R_1 \Rightarrow \begin{bmatrix}1 & 2 & 1 & 2 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 1 & 0 \\ 3 & 7 & 5 & 6\end{bmatrix}$$ - Subtract 3 times row 1 from row 4: $$R_4 = R_4 - 3R_1 \Rightarrow \begin{bmatrix}1 & 2 & 1 & 2 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0\end{bmatrix}$$ - Subtract row 2 from row 4: $$R_4 = R_4 - R_2 \Rightarrow \begin{bmatrix}1 & 2 & 1 & 2 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & -2\end{bmatrix}$$ - Subtract row 3 from row 4: $$R_4 = R_4 - R_3 \Rightarrow \begin{bmatrix}1 & 2 & 1 & 2 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2\end{bmatrix}$$ 4. **Result:** All four rows are non-zero, so the rank is 4. **Final answer:** $$\boxed{4}$$