1. The problem asks to compute the matrix $P_2$ and express its elements as fractions. 2. Typically, $P_2$ refers to the projection matrix onto the column space of a matrix $A$ with 2 columns, given by the formula: $$P_2 = A(A^T A)^{-1} A^T$$ 3. To compute $P_2$, you need the matrix $A$. Since it is not provided, let's assume $A$ is a matrix with 2 columns. 4. The steps to compute $P_2$ are: - Compute $A^T A$. - Find the inverse $(A^T A)^{-1}$. - Multiply $A$ by $(A^T A)^{-1}$ and then by $A^T$. 5. Each element of $P_2$ will be a fraction because the inverse and multiplication of matrices with integer or rational entries yield rational entries. 6. If you provide the matrix $A$, I can compute $P_2$ explicitly with fractional elements. Please provide the matrix $A$ or more details to proceed with the exact computation.