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:

\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.