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

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the product of two 3x3 matrices.
2. Step 2: Recall the matrix multiplication rule. The element in the ith row and jth column of the product matrix is the dot product of the ith row of the first matrix and the jth column of the second matrix.
3. Step 3: Set up the multiplication. For each element in the resulting matrix, multiply corresponding elements from the row of the first matrix and the column of the second matrix, then sum these products.
4. Step 4: Perform the multiplication for each element. For example, the element in the first row, first column of the product matrix is calculated by multiplying the elements of the first row of the first matrix with the first column of the second matrix and summing the results.
5. Step 5: Repeat the process for all elements in the resulting matrix to complete the matrix multiplication.

Key Concepts

Matrix Multiplication

Matrix multiplication involves taking two matrices and producing a new matrix by multiplying rows of the first matrix by columns of the second. For two matrices A (m x n) and B (n x p), the resulting matrix C (m x p) is formed where each element C[i][j] is the dot product of the i-th row of A and the j-th column of B.

Dimensions of Matrices

The dimensions of a matrix are defined by the number of rows and columns it contains, expressed as 'rows x columns'. For example, a 3x3 matrix has 3 rows and 3 columns. When multiplying matrices, the number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be valid.

Dot Product

The dot product is a fundamental operation in linear algebra that combines two equal-length sequences of numbers (usually coordinate vectors) into a single number. In the context of matrix multiplication, the dot product is used to calculate each element of the resulting matrix by summing the products of corresponding entries from the row of the first matrix and the column of the second matrix.