Suppose $$ A $$ is a symmetric matrix. Some of the entries of $$ A^{T} A $$ are shown. Find the remaining entries. $$ \left(\begin{array}{ccc}16 & \square & 7 \ 19 & 3 & 8\end{array} ight) $$

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Suppose $$ A $$ is a symmetric matrix. Some of the entries of $$ A^{T} A $$ are shown. Find the remaining entries. $$ \left(\begin{array}{ccc}16 & \square & 7 \ 19 & 3 & 8\end{array}ight) $$

UpStudy AI · Accepted Answer

To find the remaining entries of the matrix $$ A^T A $$, we need to understand the properties of symmetric matrices and the multiplication of matrices.

Given that $$ A $$ is a symmetric matrix, the product $$ A^T A $$ will also be symmetric. This means that the entries of the matrix $$ A^T A $$ satisfy the property $$ (A^T A)_{ij} = (A^T A)_{ji} $$.

The partially filled matrix is:

$$
\left(\begin{array}{ccc}
16 & \square & 7 \
19 & 3 & 8
\end{array}\right)
$$

From the symmetry property, we can deduce the following:

1. The entry in the first row and second column (denoted as $$ \square $$) must equal the entry in the second row and first column, which is $$ 19 $$. Therefore, $$ \square = 19 $$.

2. The entry in the first row and third column is $$ 7 $$, and the entry in the third row and first column must also be $$ 7 $$.

3. The entry in the second row and third column is $$ 8 $$, and the entry in the third row and second column must also be $$ 8 $$.

Now, we can fill in the matrix with the known values:

$$
\left(\begin{array}{ccc}
16 & 19 & 7 \
19 & 3 & 8 \
7 & 8 & \square
\end{array}\right)
$$

To find the value of $$ \square $$, we need to consider the properties of $$ A^T A $$. The diagonal entries of $$ A^T A $$ are the dot products of the columns of $$ A $$ with themselves. However, without additional information about the matrix $$ A $$, we cannot determine the exact value of $$ \square $$.

Thus, the completed matrix with known entries is:

$$
\left(\begin{array}{ccc}
16 & 19 & 7 \
19 & 3 & 8 \
7 & 8 & \square
\end{array}\right)
$$

The remaining entry $$ \square $$ cannot be determined without further information about the matrix $$ A $$.
The completed matrix with known entries is: $$ \left(\begin{array}{ccc} 16 & 19 & 7 \ 19 & 3 & 8 \ 7 & 8 & \square \end{array}\right) $$ The remaining entry $$ \square $$ cannot be determined with the given information.

Suppose is a symmetric matrix.
Some of the entries of are shown. Find the remaining entries.

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