Answer each question. What is the product [3x3 matrix] [3x3 matrix]?

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 \( B \). Each matrix has 3 rows and 3 columns.
2. Recall that the product of two matrices \( A \) and \( B \), where both are 3x3, is another 3x3 matrix \( C \). 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 then summing those products.
3. Write the formula for each element \( c_{ij} \) of the product matrix \( C \): \[ c_{ij} = a_{i1} \times b_{1j} + a_{i2} \times b_{2j} + a_{i3} \times b_{3j} \] where \( a_{ik} \) are elements of matrix \( A \) and \( b_{kj} \) are elements of matrix \( B \).
4. Calculate each element of matrix \( C \) by applying the formula for all \( i = 1, 2, 3 \) (rows) and \( j = 1, 2, 3 \) (columns). This means you will perform 9 separate calculations, one for each element of the resulting matrix.
5. After computing all elements \( c_{ij} \), arrange them in a 3x3 matrix format to get the final product matrix \( C = A \times B \).

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 but not commutative, meaning the order matters. Understanding this helps avoid errors when multiplying matrices and interpreting the results, especially in applications like transformations or systems of equations.