2x2 Matrix Addition and Subtraction

What are 2x2 Matrix Addition and Subtraction?

Matrix addition and subtraction are fundamental operations in linear algebra. For 2x2 matrices, these operations involve adding or subtracting corresponding elements of two matrices to produce a new 2x2 matrix. These operations are crucial in various fields, including computer graphics, physics, and data analysis.

The Formulas

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.