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

Question

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

UpStudy AI · Accepted Answer

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:**

$$
A^{T}A=\begin{pmatrix}
4 & 37 \
37 & 681
\end{pmatrix}, \quad
A^{T}\mathbf{b}=\begin{pmatrix}
854 \
12476.8
\end{pmatrix}.
$$
$$ A^{T}A = \begin{pmatrix} 4 & 37 \ 37 & 681 \end{pmatrix}, \quad A^{T}\mathbf{b} = \begin{pmatrix} 854 \ 12476.8 \end{pmatrix} $$

Calculate and , given and .

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