4x4 Matrix Addition and Subtraction

What is Matrix Addition and Subtraction?

Matrix addition and subtraction are fundamental operations in linear algebra. For two matrices A and B of the same size, their sum (A + B) or difference (A - B) is obtained by adding or subtracting corresponding elements.

The Matrix Addition and Subtraction Formulas

For two 4x4 matrices A and B, their sum C = A + B or difference C = A - B is defined as:

\[C_{ij} = A_{ij} \pm B_{ij}\]

Where:

• \(C_{ij}\) is the element in the i-th row and j-th column of the resulting matrix C
• \(A_{ij}\) is the element in the i-th row and j-th column of matrix A
• \(B_{ij}\) is the element in the i-th row and j-th column of matrix B
• \(\pm\) represents + for addition or - for subtraction

Calculation Steps

To add or subtract two 4x4 matrices:

1. Ensure both matrices are of the same size (4x4 in this case)
2. For each corresponding element in A and B:
   • Add the elements for matrix addition
   • Subtract the element of B from A for matrix subtraction
3. Place the result in the corresponding position of the resulting matrix C
4. Repeat this process for all 16 elements of the resulting matrix

Example Calculation

Let's add matrix A to matrix B:

\[A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{bmatrix}, B = \begin{bmatrix} 16 & 15 & 14 & 13 \\ 12 & 11 & 10 & 9 \\ 8 & 7 & 6 & 5 \\ 4 & 3 & 2 & 1 \end{bmatrix}\]

Calculating the first element of C:

\[C_{11} = A_{11} + B_{11} = 1 + 16 = 17\]

Continuing this process for all elements, we get:

\[C = A + B = \begin{bmatrix} 17 & 17 & 17 & 17 \\ 17 & 17 & 17 & 17 \\ 17 & 17 & 17 & 17 \\ 17 & 17 & 17 & 17 \end{bmatrix}\]

Visual Representation

This diagram visually represents the addition of matrix A and matrix B, resulting in A + B.