Matrix Addition

What Is Matrix Addition?

Matrix addition is a fundamental operation in matrix algebra where two or more matrices are combined to produce a new matrix. This operation can only be performed if the matrices being added have the same size or order.

The result of the addition is a new matrix of the same order, where each element is the sum of the corresponding elements (elements in the same position) from the original matrices.

Formal Definition of Matrix Addition

Two matrices, let's say matrix $A$ and matrix $B$, can be added if and only if both matrices have the same order.

Suppose matrix $A$ is of order $m \times n$ with elements $a_{ij}$ (element in the $i$-th row and $j$-th column), and matrix $B$ is also of order $m \times n$ with elements $b_{ij}$.

Then, the sum of matrix $A$ and matrix $B$, which we call matrix $C$, is written as $C = A + B$. Matrix $C$ will also be of order $m \times n$, with elements $c_{ij}$ defined as:

This means that each element in the resulting matrix is obtained by adding the elements that are in the same position from the two matrices being added.

How to Perform Matrix Addition

To add two matrices, follow these steps:

1. Ensure Same Order: Check if both matrices have the same number of rows and columns. If not, addition cannot be performed.
2. Add Corresponding Elements: Add the elements that are in the same row and column position from both matrices.
3. Form the Resultant Matrix: Arrange the sums of these elements into a new matrix of the same order.

Example of Matrix Addition

Suppose we have two matrices, $P$ and $Q$, as follows:

Both matrices are of order $3 \times 2$ (3 rows and 2 columns), so they can be added.

Then, $P+Q$ is:

Thus, the sum of matrix $P$ and matrix $Q$ is the matrix $\begin{bmatrix} 7 & 2 \\ 6 & 1 \\ 2 & 16 \end{bmatrix}$.

Matrices That Cannot Be Added

Suppose matrix $X = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and matrix $Y = \begin{bmatrix} 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix}$.

Matrix $X$ is of order $2 \times 2$, while matrix $Y$ is of order $2 \times 3$. Since their orders are different, matrix $X$ and matrix $Y$ cannot be added.

Properties of Matrix Addition

Matrix addition has several important properties, similar to the properties of addition for real numbers. Let $A$, $B$, and $C$ be matrices of the same order, and $O$ be the zero matrix (a matrix where all elements are zero) of the same order as $A$, $B$, and $C$.

1. Commutative Property: The order of matrix addition does not affect the result.

   $$A + B = B + A$$

   This means that adding matrix $A$ to $B$ will produce the same matrix as adding matrix $B$ to $A$.

2. Associative Property: The grouping in the addition of three or more matrices does not affect the result.

   $$(A + B) + C = A + (B + C)$$

   This means you can add $A$ and $B$ first, then add the result to $C$, or add $B$ and $C$ first, then add $A$ to the result. The final outcome will be the same.

3. Existence of an Identity Element (Zero Matrix): There exists a zero matrix $O$ that acts as the identity element in addition.

   $$A + O = O + A = A$$

   This means that if a matrix is added to a zero matrix (of the same order), the result is the matrix itself.

   This zero matrix plays a role similar to the number 0 in the addition of numbers.

4. Existence of an Additive Inverse (Opposite of a Matrix): Every matrix $A$ has an additive inverse, denoted as $-A$, which when added to $A$ results in the zero matrix $O$.

   $$A + (-A) = O$$

   The matrix $-A$ is a matrix where each element is the opposite (negative) of the corresponding elements of matrix $A$.

   For example, if $a_{ij}$ is an element of $A$, then $-a_{ij}$ is an element of $-A$.

Exercises

Problem 1

Given the following matrices:

Calculate $A+B$ and $B+A$. Then, determine if $A+C$ can be calculated and provide your explanation.

Problem 2

Determine the values of $x, y,$ and $z$ from the following matrix addition:

Problem 3

If $P = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix}$, determine the matrix $-P$ (the additive inverse of $P$) and prove that $P + (-P) = O$, where $O$ is the zero matrix of the same order.

Answer Key

Problem 1

Given matrices:

Addition of matrix $A$ and $B$ ($A+B$):

Addition of matrix $B$ and $A$ ($B+A$):

(Commutative property proven: $A+B = B+A$)

Addition of matrix $A$ and $C$ ($A+C$):

Cannot be calculated. Matrix $A$ is of order $2 \times 3$, while matrix $C$ is of order $2 \times 2$. Since their orders are different, the addition $A+C$ cannot be performed.

Problem 2

Given the matrix addition:

Perform matrix addition on the left side:

Based on the equality of two matrices, corresponding elements must be equal:

For the element in row 1, column 1: $2x+4 = 10$

For the element in row 1, column 2: $3 = 3$ (already consistent).

For the element in row 2, column 1: $-1+z = 6$

For the element in row 2, column 2: $3y+7 = 1$

Thus, the values are $x=3$, $y=-2$, and $z=7$.

Problem 3

Given matrix $P = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix}$.

The additive inverse of $P$, which is $-P$, is:

Proof that $P + (-P) = O$:

The result is the zero matrix $O$ of order $2 \times 2$. Proven.