If A = [[1,0,4], [-3,4,2], [1,-2,0]] (3x3 matrice), what is the value of AA^t

Given the matrix \( A = \begin{bmatrix} 1 & 0 & 4 \\ -3 & 4 & 2 \\ 1 & -2 & 0 \end{bmatrix} \), the problem asks to calculate \( AA^t \), where \( A^t \) is the transpose of matrix \( A \).

### Step 1: Transpose of Matrix \( A \)
The transpose of a matrix \( A \), denoted by \( A^t \), is obtained by swapping rows with columns. For matrix \( A \):

\[
A^t = \begin{bmatrix} 
1 & -3 & 1 \\ 
0 & 4 & -2 \\ 
4 & 2 & 0 
\end{bmatrix}
\]

### Step 2: Matrix Multiplication of \( A \) and \( A^t \)
Now, we calculate \( AA^t \) by multiplying matrix \( A \) by its transpose \( A^t \).

\[
AA^t = A \cdot A^t = \begin{bmatrix} 
1 & 0 & 4 \\ 
-3 & 4 & 2 \\ 
1 & -2 & 0 
\end{bmatrix} \cdot \begin{bmatrix} 
1 & -3 & 1 \\ 
0 & 4 & -2 \\ 
4 & 2 & 0 
\end{bmatrix}
\]

We multiply the matrices by using the formula for matrix multiplication:

\[
(AA^t)_{ij} = \sum_{k=1}^n A_{ik} \cdot (A^t)_{kj}
\]

### Performing the calculations for each element of \( AA^t \):

- First row:
  \[
  \begin{aligned}
  (AA^t)_{11} &= 1 \cdot 1 + 0 \cdot 0 + 4 \cdot 4 = 1 + 0 + 16 = 17 \\
  (AA^t)_{12} &= 1 \cdot (-3) + 0 \cdot 4 + 4 \cdot 2 = -3 + 0 + 8 = 5 \\
  (AA^t)_{13} &= 1 \cdot 1 + 0 \cdot (-2) + 4 \cdot 0 = 1 + 0 + 0 = 1
  \end{aligned}
  \]

- Second row:
  \[
  \begin{aligned}
  (AA^t)_{21} &= -3 \cdot 1 + 4 \cdot 0 + 2 \cdot 4 = -3 + 0 + 8 = 5 \\
  (AA^t)_{22} &= -3 \cdot (-3) + 4 \cdot 4 + 2 \cdot 2 = 9 + 16 + 4 = 29 \\
  (AA^t)_{23} &= -3 \cdot 1 + 4 \cdot (-2) + 2 \cdot 0 = -3 - 8 + 0 = -11
  \end{aligned}
  \]

- Third row:
  \[
  \begin{aligned}
  (AA^t)_{31} &= 1 \cdot 1 + (-2) \cdot 0 + 0 \cdot 4 = 1 + 0 + 0 = 1 \\
  (AA^t)_{32} &= 1 \cdot (-3) + (-2) \cdot 4 + 0 \cdot 2 = -3 - 8 + 0 = -11 \\
  (AA^t)_{33} &= 1 \cdot 1 + (-2) \cdot (-2) + 0 \cdot 0 = 1 + 4 + 0 = 5
  \end{aligned}
  \]

### Step 3: Resultant Matrix
Thus, the result of \( AA^t \) is:

\[
AA^t = \begin{bmatrix} 
17 & 5 & 1 \\ 
5 & 29 & -11 \\ 
1 & -11 & 5 
\end{bmatrix}
\]

Would you like to go over any specific step or need further explanation?

### Related Questions:
1. How do you calculate the determinant of a matrix?
2. What is the significance of a matrix's transpose?
3. How can matrix multiplication be optimized for large matrices?
4. What are the properties of symmetric matrices?
5. How is matrix multiplication used in solving systems of equations?

### Tip:
When multiplying matrices, always verify if the inner dimensions match to ensure the operation is valid.

If A = [[1,0,4], [-3,4,2], [1,-2,0]] (3x3 matrice), what is the value of AA^t?

This solution demonstrates how to calculate the product of a matrix and its transpose (AA^t) for the matrix A = [[1,0,4], [-3,4,2], [1,-2,0]]. The detailed steps cover key concepts like matrix multiplication and transpose, suitable for undergraduate linear algebra courses.

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