2x2 Matrix Multiplication

What is 2x2 Matrix Multiplication?

Matrix multiplication is a fundamental operation in linear algebra. For 2x2 matrices, it involves multiplying two 2x2 matrices to produce a new 2x2 matrix. This operation is crucial in various fields, including computer graphics, physics, and data analysis.

The Formula

Given two 2x2 matrices A and B:

\[A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}\]

The product C = AB is calculated as:

\[C = AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}\]

Calculation Steps

1. Multiply the elements of the first row of A with the corresponding elements of the first column of B and sum the results to get c₁₁.
2. Multiply the elements of the first row of A with the corresponding elements of the second column of B and sum the results to get c₁₂.
3. Multiply the elements of the second row of A with the corresponding elements of the first column of B and sum the results to get c₂₁.
4. Multiply the elements of the second row of A with the corresponding elements of the second column of B and sum the results to get c₂₂.

Example

Let's multiply these two matrices:

\[A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}\]

Step-by-step calculation:

1. \(c_{11} = (2 \times 5) + (3 \times 7) = 10 + 21 = 31\)
2. \(c_{12} = (2 \times 6) + (3 \times 8) = 12 + 24 = 36\)
3. \(c_{21} = (1 \times 5) + (4 \times 7) = 5 + 28 = 33\)
4. \(c_{22} = (1 \times 6) + (4 \times 8) = 6 + 32 = 38\)

The result is:

\[C = AB = \begin{bmatrix} 31 & 36 \\ 33 & 38 \end{bmatrix}\]

Visual Representation

This diagram illustrates the multiplication of matrices A and B to produce the result matrix C.