Calculate \( A^{T} A \) and \( A^{T} \mathbf{b} \), given \( A=\left[\begin{array}{cc}1 & 2 \\ 1 & 25 \\ 1 & 6 \\ 1 & 4\end{array}\right] \) and \( \mathbf{b}=\left[\begin{array}{c}126 \\ 428.4 \\ 158.2 \\ 141.4\end{array}\right] \). \( A^{T} A=\left[\begin{array}{ll}\text { Ex: } 1 \\ \square & \square\end{array}\right] \) \( A^{T} \mathbf{b}=\left[\begin{array}{l}\text { Ex: } 1.2 \\ \square\end{array}\right. \)

Let \[ A=\begin{pmatrix} 1 & 2 \\ 1 & 25 \\ 1 & 6 \\ 1 & 4 \end{pmatrix}, \quad \mathbf{b}=\begin{pmatrix} 126 \\ 428.4 \\ 158.2 \\ 141.4 \end{pmatrix}. \] **Step 1. Compute \(A^{T} A\):** The transpose of \(A\) is \[ A^{T}=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 2 & 25 & 6 & 4 \end{pmatrix}. \] Then, \[ A^{T}A=\begin{pmatrix} \sum_{i=1}^4 A_{i1}^2 & \sum_{i=1}^4 A_{i1}A_{i2} \\ \sum_{i=1}^4 A_{i1}A_{i2} & \sum_{i=1}^4 A_{i2}^2 \end{pmatrix}. \] - For the \((1,1)\) entry: \[ \sum_{i=1}^4 A_{i1}^2 = 1^2 + 1^2 + 1^2 + 1^2 = 4. \] - For the \((1,2)\) and \((2,1)\) entry: \[ \sum_{i=1}^4 A_{i1}A_{i2} = 1\cdot2 + 1\cdot25 + 1\cdot6 + 1\cdot4 = 2 + 25 + 6 + 4 = 37. \] - For the \((2,2)\) entry: \[ \sum_{i=1}^4 A_{i2}^2 = 2^2 + 25^2 + 6^2 + 4^2 = 4 + 625 + 36 + 16 = 681. \] Thus, \[ A^{T}A=\begin{pmatrix} 4 & 37 \\ 37 & 681 \end{pmatrix}. \] **Step 2. Compute \(A^{T}\mathbf{b}\):** The product \(A^{T}\mathbf{b}\) is a \(2 \times 1\) vector given by \[ A^{T}\mathbf{b}=\begin{pmatrix} \sum_{i=1}^4 A_{i1}\,b_{i} \\ \sum_{i=1}^4 A_{i2}\,b_{i} \end{pmatrix}. \] - For the first entry: \[ \sum_{i=1}^4 A_{i1}\,b_{i} = 1\cdot126 + 1\cdot428.4 + 1\cdot158.2 + 1\cdot141.4 = 126 + 428.4 + 158.2 + 141.4 = 854. \] - For the second entry: \[ \sum_{i=1}^4 A_{i2}\,b_{i} = 2\cdot126 + 25\cdot428.4 + 6\cdot158.2 + 4\cdot141.4. \] Calculate each term: \[ 2\cdot126 = 252, \] \[ 25\cdot428.4 = 10710 \quad (\text{since } 428.4 \cdot 25 = 428.4 \cdot (20+5)=8568+2142), \] \[ 6\cdot158.2 = 949.2, \] \[ 4\cdot141.4 = 565.6. \] Now sum these: \[ 252 + 10710 = 10962, \] \[ 10962 + 949.2 = 11911.2, \] \[ 11911.2 + 565.6 = 12476.8. \] Thus, \[ A^{T}\mathbf{b}=\begin{pmatrix} 854 \\ 12476.8 \end{pmatrix}. \] **Final Answers:** \[ A^{T}A=\begin{pmatrix} 4 & 37 \\ 37 & 681 \end{pmatrix}, \quad A^{T}\mathbf{b}=\begin{pmatrix} 854 \\ 12476.8 \end{pmatrix}. \]