Matrix multiplication is a fundamental operation in linear algebra, particularly when dealing with matrices of different dimensions. This article will focus on the multiplication of a 2×3 matrix by another matrix, exploring the method and providing examples to enhance understanding.

Understanding 2×3 Matrix

A 2×3 matrix consists of 2 rows and 3 columns. For instance, let A be a 2×3 matrix represented as:

A = [a11, a12, a13]

[a21, a22, a23]

Multiplying a 2×3 Matrix by Another Matrix

When multiplying a 2×3 matrix with a 3×2 matrix, the result is a 2×2 matrix. Let’s denote the 3×2 matrix as B:

B = [b11, b12]

[b21, b22]

[b31, b32]

The resulting matrix C will be a 2×2 matrix where each element cij is calculated by taking the dot product of the rows of matrix A and the columns of matrix B.

Example and Calculation

To illustrate, assume matrices A and B are:

A = [1, 2, 3]

[4, 5, 6]

B = [7, 8]

[9, 10]

[11, 12]

The product C = A × B is computed as:

C11 = (17 + 29 + 311) = 58

C12 = (18 + 210 + 312) = 64

C21 = (47 + 59 + 611) = 139

C22 = (48 + 510 + 612) = 154

Thus, matrix C is:

C = [58, 64]

[139, 154]

In summary, multiplying a 2×3 matrix with a 3×2 matrix follows a specific method that results in a 2×2 matrix. Understanding this process is crucial for applications in various fields such as computer graphics and data analysis.