Solve the following matrix A superscript T.A, knowing. AsuperscriptT = [a,b;c,d;e,f], A= [x,y,z; p,q,R]

To solve the matrix multiplication \( A^T A \), we need to understand the matrices involved:

Given matrices:
1. \( A^T \) (transpose of \( A \)) is given by:
   \[
   A^T = \begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix}
   \]

2. \( A \) is given by:
   \[
   A = \begin{bmatrix} x & y & z \\ p & q & r \end{bmatrix}
   \]

### Step 1: Matrix Multiplication of \( A^T \) and \( A \)

The resulting matrix \( A^T A \) will be a \( 3 \times 3 \) matrix since \( A^T \) is \( 3 \times 2 \) and \( A \) is \( 2 \times 3 \).

Let's multiply these matrices:

\[
A^T A = 
\begin{bmatrix} 
a & b \\ 
c & d \\ 
e & f 
\end{bmatrix} 
\times 
\begin{bmatrix} 
x & y & z \\ 
p & q & r 
\end{bmatrix}
\]

### Step 2: Compute Each Element of the Resulting Matrix

1. **Element (1,1):**
   \[
   (a \cdot x) + (b \cdot p)
   \]

2. **Element (1,2):**
   \[
   (a \cdot y) + (b \cdot q)
   \]

3. **Element (1,3):**
   \[
   (a \cdot z) + (b \cdot r)
   \]

4. **Element (2,1):**
   \[
   (c \cdot x) + (d \cdot p)
   \]

5. **Element (2,2):**
   \[
   (c \cdot y) + (d \cdot q)
   \]

6. **Element (2,3):**
   \[
   (c \cdot z) + (d \cdot r)
   \]

7. **Element (3,1):**
   \[
   (e \cdot x) + (f \cdot p)
   \]

8. **Element (3,2):**
   \[
   (e \cdot y) + (f \cdot q)
   \]

9. **Element (3,3):**
   \[
   (e \cdot z) + (f \cdot r)
   \]

### Final Result

Putting it all together, the matrix \( A^T A \) is:

\[
A^T A = 
\begin{bmatrix} 
ax + bp & ay + bq & az + br \\ 
cx + dp & cy + dq & cz + dr \\ 
ex + fp & ey + fq & ez + fr 
\end{bmatrix}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you compute the transpose of a matrix?
2. What are the properties of matrix multiplication?
3. How can you find the determinant of a \( 3 \times 3 \) matrix?
4. What is the significance of the product \( A^T A \) in linear algebra?
5. How can we compute the inverse of a matrix?

**Tip:** Always ensure the dimensions match correctly when performing matrix multiplication; the number of columns in the first matrix must equal the number of rows in the second.

Learn how to solve the matrix multiplication A transpose times A with a step-by-step explanation. Understand the concepts of matrix algebra, transposition, and multiplication involved in this problem. Suitable for advanced learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation