Find ATA and AAT.
A =

1	−1

5	2
0	−6
(a)    ATA

(b)    AAT

Which of the products are symmetric? (Select all that apply.)
ATA
AAT

Given the matrix \( A \):

\[
A = \begin{pmatrix} 
1 & -1 \\
5 & 2 \\
0 & -6 
\end{pmatrix}
\]

We need to calculate \( A^T A \) and \( A A^T \) and determine which of these products are symmetric.

### (a) \( A^T A \)

1. First, find the transpose of \( A \), denoted as \( A^T \):

\[
A^T = \begin{pmatrix} 
1 & 5 & 0 \\
-1 & 2 & -6 
\end{pmatrix}
\]

2. Multiply \( A^T \) by \( A \):

\[
A^T A = \begin{pmatrix} 
1 & 5 & 0 \\
-1 & 2 & -6 
\end{pmatrix} \times \begin{pmatrix} 
1 & -1 \\
5 & 2 \\
0 & -6 
\end{pmatrix}
\]

\[
A^T A = \begin{pmatrix} 
1(1) + 5(5) + 0(0) & 1(-1) + 5(2) + 0(-6) \\
-1(1) + 2(5) + -6(0) & -1(-1) + 2(2) + -6(-6) 
\end{pmatrix}
\]

\[
A^T A = \begin{pmatrix} 
1 + 25 + 0 & -1 + 10 + 0 \\
-1 + 10 + 0 & 1 + 4 + 36 
\end{pmatrix}
\]

\[
A^T A = \begin{pmatrix} 
26 & 9 \\
9 & 41 
\end{pmatrix}
\]

### (b) \( A A^T \)

1. Now multiply \( A \) by \( A^T \):

\[
A A^T = \begin{pmatrix} 
1 & -1 \\
5 & 2 \\
0 & -6 
\end{pmatrix} \times \begin{pmatrix} 
1 & 5 & 0 \\
-1 & 2 & -6 
\end{pmatrix}
\]

\[
A A^T = \begin{pmatrix} 
1(1) + (-1)(-1) & 1(5) + (-1)(2) & 1(0) + (-1)(-6) \\
5(1) + 2(-1) & 5(5) + 2(2) & 5(0) + 2(-6) \\
0(1) + (-6)(-1) & 0(5) + (-6)(2) & 0(0) + (-6)(-6)
\end{pmatrix}
\]

\[
A A^T = \begin{pmatrix} 
1 + 1 & 5 - 2 & 0 + 6 \\
5 - 2 & 25 + 4 & 0 - 12 \\
0 + 6 & 0 - 12 & 0 + 36
\end{pmatrix}
\]

\[
A A^T = \begin{pmatrix} 
2 & 3 & 6 \\
3 & 29 & -12 \\
6 & -12 & 36 
\end{pmatrix}
\]

### Symmetry Check

- A matrix is symmetric if it is equal to its transpose, i.e., \( M = M^T \).

For \( A^T A \):

\[
A^T A = \begin{pmatrix} 
26 & 9 \\
9 & 41 
\end{pmatrix}
\]

\( A^T A \) is symmetric because \( A^T A = (A^T A)^T \).

For \( A A^T \):

\[
A A^T = \begin{pmatrix} 
2 & 3 & 6 \\
3 & 29 & -12 \\
6 & -12 & 36 
\end{pmatrix}
\]

\( A A^T \) is symmetric because \( A A^T = (A A^T)^T \).

### Conclusion

Both \( A^T A \) and \( A A^T \) are symmetric matrices.

---

Would you like more details or have any questions?

Here are 8 related questions to consider:

1. How do you verify if a matrix is symmetric?
2. What is the significance of the transpose of a matrix?
3. Can \( A^T A \) and \( A A^T \) be different sizes?
4. How would the result change if \( A \) were a square matrix?
5. What are the properties of symmetric matrices?
6. Why are \( A^T A \) and \( A A^T \) always symmetric for any matrix \( A \)?
7. How do symmetric matrices relate to eigenvalues and eigenvectors?
8. What is the determinant of a symmetric matrix?

**Tip:** Always remember that the product \( A^T A \) and \( A A^T \) for any real matrix \( A \) will always be symmetric.

Explore the properties of matrix products ATA and AAT. Understand how to calculate these products and determine their symmetry. Detailed step-by-step solutions provided.

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