Answer each question. What is the product ?

Verified step by step guidance

1. Identify the two 3x3 matrices given in the problem. Let's call the first matrix \( A \) and the second matrix \( B \).
2. Recall that the product of two matrices \( A \) and \( B \), where both are 3x3, is another 3x3 matrix \( C = AB \). Each element \( c_{ij} \) of matrix \( C \) is found by multiplying the elements of the \( i \)-th row of \( A \) by the corresponding elements of the \( j \)-th column of \( B \) and summing the results.
3. Write the formula for each element \( c_{ij} \) of the product matrix \( C \): \[ c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + a_{i3}b_{3j} \]
4. For each element in the resulting matrix \( C \), perform the multiplication and addition as per the formula above. Do this for all rows \( i = 1, 2, 3 \) and columns \( j = 1, 2, 3 \).
5. After calculating all nine elements \( c_{ij} \), arrange them into the 3x3 matrix \( C \), which is the product of the two given matrices.

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. For two 3x3 matrices, each element in the product matrix is calculated by taking the dot product of a row from the first matrix and a column from the second matrix.

Dimensions and Compatibility

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. Since both matrices are 3x3, they are compatible for multiplication, and the resulting matrix will also be 3x3.

Properties of Matrix Multiplication

Matrix multiplication is associative and distributive but not commutative, meaning AB does not necessarily equal BA. Understanding these properties helps in correctly performing and interpreting matrix products.