Matrix multiplication is a fundamental operation in linear algebra, especially when dealing with matrices of different dimensions. When multiplying a 3×2 matrix by a 2×2 matrix, it’s essential to understand the rules and process involved. This article provides a detailed guide on performing this type of matrix multiplication, including step-by-step instructions and examples.

Understanding Matrix Dimensions

To start, let’s clarify the dimensions of the matrices involved. A 3×2 matrix has 3 rows and 2 columns, while a 2×2 matrix has 2 rows and 2 columns. For matrix multiplication to be valid, the number of columns in the first matrix must match the number of rows in the second matrix. Here, the multiplication is feasible because the 2 columns of the first matrix align with the 2 rows of the second matrix.

Step-by-Step Multiplication Process

To multiply a 3×2 matrix (A) with a 2×2 matrix (B), follow these steps:

1. Take each row of the 3×2 matrix and multiply it by each column of the 2×2 matrix.

2. Sum the products for each pair of row and column to get the elements of the resulting matrix.

3. The resulting matrix will have the dimensions of 3×2, matching the number of rows from the first matrix and columns from the second matrix.

Example and Result

Suppose matrix A is:

[[a11, a12],

[a21, a22],

[a31, a32]]

And matrix B is:

[[b11, b12],

[b21, b22]]

The resulting matrix C, after multiplication, will be:

[[a11b11 + a12b21, a11b12 + a12b22],

[a21b11 + a22b21, a21b12 + a22b22],

[a31b11 + a32b21, a31b12 + a32b22]]

Summary

Matrix multiplication involving a 3×2 matrix and a 2×2 matrix is a straightforward process as long as the dimension rules are followed. By multiplying each row of the first matrix by each column of the second matrix and summing the products, you can compute the resultant matrix effectively. This process is crucial in various applications, including computer graphics and data transformations.