# Addition of Matrices – Definition, Rules, Properties and Examples | How to do Addition of Matrices?

Matrix addition or addition of matrices is the addition operation performed between two or more matrices. The addition of matrices is different from the addition of numbers. We will follow some rules to add matrices. A matrix is a rectangular array of numbers, expressions, letters, symbols, etc. arranged in rows and columns.

We can use different addition methods like the element-wise addition of matrices and the direct sum of matrices. Also, we can perform different operations on the matrices like subtraction and multiplication. The matrix addition is only for the matrices having the same dimension or size. The 10th Grade Math Addition of Matrix is clearly explained in this article below.

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## Matrix Addition – Definition | What is the Addition of Matrices?

The addition of matrices is the addition operation performed on different matrices having the same order. The elements of the same order matrices are added individually. If A = [aij] and B = [bij] are two matrices with the same order or dimension (the same number of rows and columns), then the addition of matrices A and B becomes: A+B = [aij] + [bij] = [aij + bij]. If matrix A is a 2 Ã— 3 matrices, then the B matrix will also be a 2 Ã— 3 matrices to perform an addition operation.

### Types of Addition of Matrices

There are two different types of addition methods available for matrices. They are Element-wise Addition of Matrices and the other one is a direct sum of the matrices.

### How to do Addition of Matrices?

Addition of Matrices can be done using two different methods and they are explained below in detail for your knowledge.

In Element-wise Matrix Addition, we add the elements in each row and column to the respective elements in the row and column of the other matrix. If A and B are two matrices of the same order with m number of rows and n number of columns, then the addition becomes A + B = [aij] + [bij] = [aij + bij], where ij denotes the position of each element in the ith row and jth column. Here A = [aij] and B = [bij].

#### Direct Sum – Matrix Addition

In Direct Sum – Matrix Addition, we use the âŠ•to denote the addition operation of matrices. The order of the matrices needs to be the same while doing the Direct sum addition of matrices. If X is the order of m Ã— n (m rows and n columns) and Y is the order of p Ã— q (p rows and q columns). Then, the addition becomes XâŠ•Y is (m+p) Ã— (n+q). The direct sum of matrices is associative, i.e, (XâŠ•Y)âŠ•Z = XâŠ•(YâŠ•Z).

Properties of Addition of Matrices are given below. Check out all the properties and apply them while solving addition problems.

Commutative Property: The addition of two matrices is commutative when it obeys A + B = B +A where A = [aij] and B = [bij] are two matrices of the same order, say m x n.
Associative Property: We can say the matrix addition is associative when it obeys (A + B) + C = A + (B + C) where If A = [aij], B = [bij] and C = [cij] are three matrices of order m Ã— n.
Existence of Additive Identity: The Additive Identity of matrix A = [aij] of order m Ã— n is the A + O = O + A = A where O is a zero matrix of order m Ã— n. Here O matrix is the additive identity for matrix addition.
Existence of Additive Inverse: The Additive Inverse of matrix A = [aij] of order m Ã— n is the A + (-A) = O = A + (-A) where -A = [-aij] is a inverse of A of order m Ã— n. Here -A matrix is the additive inverse for matrix addition.
Transpose Property: The transpose of the sum of two matrices (A + B) is equal to the sum of the transposes of the respective matrices ATÂ + BT. (A + B)TÂ = ATÂ + BT
Determinant Property: The determinant of the sum of two matrices |A + B| is equal to the sum of the determinants of the respective matrices |A| + |B|. |A + B| = |A| + |B|

The below examples are solved with different tips and tricks. All the properties are used to solve the matrix addition problems. So, try to practice every problem to know how to solve the addition of matrices problems.

Example 1.
$$A =\left[ \begin{matrix} 12&11 \cr 42&51 \cr \end{matrix} \right]$$ and $$B =\left[ \begin{matrix} 6&11 \cr \end{matrix} \right]$$

Solution:
Given matrices are $$A =\left[ \begin{matrix} 12&11 \cr 42&51 \cr \end{matrix} \right]$$ and $$B =\left[ \begin{matrix} 6&11 \cr \end{matrix} \right]$$
The matrix A order is 2 Ã— 2 and the matrix B order is 1 Ã— 2. To perform an addition operation between two matrices, the order of both matrices must be the same. The order of the given matrices is not the same.

Therefore, we cannot add given matrices as their order is not the same.

Example 2.
Write the elements of the sum matrix R = P + Q explicitly by addition of matrices P and Q of dimension 1 Ã— 2 whose elements are given as: p11 = 2, p12 = 8 and qb11 = -2, q12 = -16.

Solution:
Given that P and Q are of the same order of 1 Ã— 2. Therefore, we can perform the addition operation for P and Q. As the P and Q order is 1 Ã— 2, the output will also have the same order of 1 Ã— 2. By adding the corresponding elements,
r11 = p11 + q11 = 2 + (-2) = 0
r12 = p12 + q12 = 8 + (-16) = -8

Therefore, the answer is r11 = 0 and r12 = -8.

Example 3.
Determine the element of the second row and third column of the matrix X + Y using the addition of matrices definition if x23 = -19 is an element of X and y23 = 30 is an element in Y.

Solution:
Given that x23 = -19 is an element of X and y23 = 30 is an element in Y.
We need to add x23 and y23 to find the element of the second row and third column of the matrix X + Y.
x23 + y23 = -19 + 30 = 11.

Therefore, the element in the second row and third column of A + B is 11.

Example 4.
Add 2 Ã— 2 order matrices given below.
$$A =\left[ \begin{matrix} 8&10 \cr 40&36 \cr \end{matrix} \right]$$ and $$B =\left[ \begin{matrix} 6&12 \cr 4&8 \cr \end{matrix} \right]$$

Solution:
Given matrices are $$A =\left[ \begin{matrix} 8&10 \cr 40&36 \cr \end{matrix} \right]$$ and $$B =\left[ \begin{matrix} 6&12 \cr 4&8 \cr \end{matrix} \right]$$
Now, add the elements of A with the elements of B.
$$\left[ \begin{matrix} 8&10 \cr 40&36 \cr \end{matrix} \right]$$ + $$\left[ \begin{matrix} 6&12 \cr 4&8 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 8 + 6&10 + 12 \cr 40 + 4&36 + 8 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 14&22 \cr 44&44 \cr \end{matrix} \right]$$

Therefore, the addition of matrices A and B is $$\left[ \begin{matrix} 14&22 \cr 44&44 \cr \end{matrix} \right]$$.

1. What is the Matrix Addition in Math?

The addition of matrices is the addition between two or more matrices of the same order. If A = [aij] and B = [bij], then the sum of the two matrices A and B is A+B = [aij] + [bij] = [aij + bij], where ij denotes the position of each element in ith row and jth column.

2. What is the Necessary Condition for Addition of Matrices?

To add two or more matrices, those matrices must have the same order. The number of rows and columns of the matrices must be equal to add them.

3. Is Addition of Matrices Commutative?

Yes, the addition of matrcies is commutative when A + B = B + A where A = [aij] and B = [bij] of same order are added.

4. Can we add a 2 x 2 matrix to a 3 x 3 matrix?

No, it is not possible to add a 2 x 2 matrix to a 3 x 3 matrix as their order is different.

5. Is the Matrix Addition possible for Matrices of Different Dimensions?

No, it is not possible to add matrices of Different Dimensions.