2.2. Addition of Matrices#

We can add/subtract two matrices only if all matrices involved in the addition/subtraction need to have the same size.

Matrix Addition

Assume that \(A\) and \(B\) are matrices of the same size, both having \(m\) rows and \(n\) columns. The summation of these matrices, denoted as \(A + B\), is a new matrix calculated by adding the corresponding entries of matrices \(A\) and \(B\) [Kuttler and Farah, 2020, Nicholson, 2018].

Mathematically, if \(A\) and \(B\) are \(m\times n\) matrices, then the matrix \(A + B\) is also an \(m\times n\) matrix, and its entries are obtained as follows:

\[\begin{align*} (A + B)_{ij} = A_{ij} + B_{ij} \end{align*}\]

Where:

  • \((A + B)_{ij}\) is the entry in the \(i\)-th row and \(j\)-th column of the matrix \(A + B\).

  • \(A_{ij}\) is the entry in the \(i\)-th row and \(j\)-th column of matrix \(A\).

  • \(B_{ij}\) is the entry in the \(i\)-th row and \(j\)-th column of matrix \(B\).

To put it simply, to get the entry in the resulting matrix \(A + B\) at position \((i,j)\), we add the corresponding entries from matrices \(A\) and \(B\) that are in the same position.

Example: For example, consider the following matrices:

\[\begin{align*} A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix} \end{align*}\]
\[\begin{align*} B = \begin{bmatrix} 1 & 0 \\ 2 & -1 \end{bmatrix} \end{align*}\]

The summation of matrices \(A\) and \(B\), denoted as \(A + B\), will be:

\[\begin{align*} A + B = \begin{bmatrix} 2+1 & 4+0 \\ 1+2 & 3+(-1) \end{bmatrix} = \begin{bmatrix} 3 & 4 \\ 3 & 2 \end{bmatrix} \end{align*}\]

So, the resulting matrix \(A + B\) is a \(2\times 2\) matrix with entries obtained by adding the corresponding entries of matrices \(A\) and \(B\).

Example: Let \(A=\begin{bmatrix} 1 & -2 & 4\\ 3 & 0 & 7 \end{bmatrix}\) and \(B=\begin{bmatrix} 6 & 2 & 4\\ -2 & 1 & -1 \end{bmatrix}\). Find \(A+B\).

Solution:

\[\begin{align*} A+B &=\begin{bmatrix} 1 & -2 & 4\\ 3 & 0 & 7 \end{bmatrix}+\begin{bmatrix} 6 & 2 & 4\\ -2 & 1 & -1 \end{bmatrix} \\ & =\begin{bmatrix} 1+6 & -2+2 & 4+4\\ 3+(-2) & 0+1 & 7+(-1) \end{bmatrix} \\ & =\begin{bmatrix} 7 & 0 & 8\\ 1 & 1 & 6 \end{bmatrix}. \end{align*}\]

Example: If \(C=\begin{bmatrix} 1 & -3 & 4\\ 0 & 1 & 2 \end{bmatrix}\) and \(D=\begin{bmatrix} 1 & 1\\ 1 & 3 \end{bmatrix}\), we cannot have \(C+D\) since \(C\) is \(2\times 3\) and \(D\) is \(2\times 2\).

Proposition - Properties of Matrix Addition:

Let \(A\), \(B\), and \(C\) be matrices of the same size. Then [Kuttler and Farah, 2020, Nicholson, 2018],

  • Commutative Law of Addition: \(A+B = B+A\)

  • Associative Law of Addition: \((A+B)+C = A+(B+C)\)

  • Existence of an Additive Identity: There exists a zero matrix 0 such that \(A+0 = A\)

  • Existence of an Additive Inverse: There exists a matrix \(-A\) such that \(A+(-A) = 0\)