# Addition of Two Matrices – Definition, Conditions, Properties | How to Add Two Matrices?

The addition of Two Matrices is the process of adding only two matrices. The addition operation on two matrices is possible only when they have the same order like 2 × 2 or 3 × 3. The Two Matrix Addition has some properties. We have given all the properties and problems along with some tips and tricks.

Read the entire article and learn how to add two matrices. To learn the Addition of matrices along with the Addition of Two Matrices, read the 10th Grade Math articles on our website. You can also add 3 × 3 matrices, 4 × 4 matrices, etc. If the order of the matrices is the same, then you can go through the addition process on matrices. Check out the entire article and practice every problem given here.

## What is the Addition of Two Matrices?

The addition of two matrices is the addition operation performed on two matrices. If A = [aij] and B = [bij] are two matrices with the same order or dimension having the same number of rows and same number of columns, then the addition of matrices A and B becomes: A+B = [aij] + [bij] = [aij + bij]. To add two matrices, they must have the same order.

### How to Add Two Matrices?

The below process will help you to know how to add two matrices. The addition process of matrices is easy compared to the subtraction or multiplication process.

• Firstly, check if the given matrices are having the same order or not.
• If the matrices have the same order, then check out the elements of the same position.
• Add the elements of the first matrix with the respective elements of the second matrix.
• Then finally write the output of the addition of two matrices.

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Below are the examples of Two Matrix Addition. Solve all problems on your own and check out the answers to test your preparation.

Example 1. If $$A =\left[ \begin{matrix} 12&14 \cr 2&32 \cr \end{matrix} \right]$$ and $$B =\left[ \begin{matrix} 15&20 \cr 7&9 \cr \end{matrix} \right]$$, then find the addition of A and B.

Solution:
Given matrices are $$A =\left[ \begin{matrix} 12&14 \cr 2&32 \cr \end{matrix} \right]$$ and $$B =\left[ \begin{matrix} 15&20 \cr 7&9 \cr \end{matrix} \right]$$
Both matrices have the same order of 2 × 2. Now, Add the elements of the first matrix with the respective elements of the second matrix.
$$\left[ \begin{matrix} 12&14 \cr 2&32 \cr \end{matrix} \right]$$ + $$\left[ \begin{matrix} 15&20 \cr 7&9 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 12 + 15&14 + 20 \cr 2 + 7&32 + 9 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 27&34 \cr 9&41 \cr \end{matrix} \right]$$

Therefore, the addition of A and B is $$\left[ \begin{matrix} 27&34 \cr 9&41 \cr \end{matrix} \right]$$.

Example 2.
Add X and Y where $$X =\left[ \begin{matrix} 0&1 \cr 0&1 \cr \end{matrix} \right]$$ and $$Y =\left[ \begin{matrix} 1&0 \cr 0&1 \cr \end{matrix} \right]$$

Solution:
Given matrices are $$X =\left[ \begin{matrix} 0&1 \cr 0&1 \cr \end{matrix} \right]$$ and $$Y =\left[ \begin{matrix} 1&0 \cr 0&1 \cr \end{matrix} \right]$$
Both matrices have the same order of 2 × 2. Now, Add the elements of the first matrix with the respective elements of the second matrix.
$$\left[ \begin{matrix} 0&1 \cr 0&1 \cr \end{matrix} \right]$$ + $$\left[ \begin{matrix} 1&0 \cr 0&1 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 0 + 1&1 + 0 \cr 0 + 0&1 + 1 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 1&1 \cr 0&2 \cr \end{matrix} \right]$$

Therefore, the addition of X and Y is $$\left[ \begin{matrix} 1&1 \cr 0&2 \cr \end{matrix} \right]$$

Example 3.
Find a and b where the addition of $$A = \left[ \begin{matrix} 3 + a&1 \cr b + 4&2 \cr \end{matrix} \right]$$ and $$B = \left[ \begin{matrix} 4&1 \cr 2&4 \cr \end{matrix} \right]$$ is $$\left[ \begin{matrix} 12&16 \cr 14&10 \cr \end{matrix} \right]$$

Solution:
Given that $$A = \left[ \begin{matrix} 3 + a&1 \cr b + 4&2 \cr \end{matrix} \right]$$ and $$B = \left[ \begin{matrix} 4&1 \cr 2&4 \cr \end{matrix} \right]$$ is $$\left[ \begin{matrix} 12&16 \cr 14&10 \cr \end{matrix} \right]$$
Now, add the elements of A and B.
$$A + B = \left[ \begin{matrix} 3 + a + 4&1 + 1 \cr b + 4 + 2&2 + 4 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 12&16 \cr 14&10 \cr \end{matrix} \right]$$
So, a + 7 = 12; a = 12 – 7; a = 5.
b+ 6 = 14; b = 14 – 6; b = 8.

Therefore, the values of a and b are a = 5, and b = 8.

Example 4.
Add $$\left[ \begin{matrix} 2&5&7 \cr 9&10&11 \cr \end{matrix} \right]$$ and $$\left[ \begin{matrix} 3&6&8 \cr 14&15&12 \cr \end{matrix} \right]$$

Solution:
Given matrices are $$\left[ \begin{matrix} 2&5&7 \cr 9&10&11 \cr \end{matrix} \right]$$ and $$\left[ \begin{matrix} 3&6&8 \cr 14&15&12 \cr \end{matrix} \right]$$
Given matrices are 2 × 3 matrices of the same order. Now, Add the elements of the first matrix with the respective elements of the second matrix.
$$\left[ \begin{matrix} 2&5&7 \cr 9&10&11 \cr \end{matrix} \right]$$ + $$\left[ \begin{matrix} 3&6&8 \cr 14&15&12 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 2 + 3&5 + 6&7 + 8 \cr 9 + 14&10 + 15&11 + 12 \cr \end{matrix} \right]$$ = $$\left[ \begin{matrix} 5&11&15 \cr 23&25&23 \cr \end{matrix} \right]$$

Therefore, the required matrix is $$\left[ \begin{matrix} 5&11&15 \cr 23&25&23 \cr \end{matrix} \right]$$

### FAQs on Addition of Two Matrices

1. What is the Addition of Two Matrices?

The Addition of Two Matrices is the addition of two matrices of the same dimensions or order.

2. Is it possible to add 1 × 2 and 2 × 1 matrix?

No, it is not possible to add 1 × 2 and 2 × 1 matrices.

3. What is the rule for matrix addition?

The main rule for adding matrices is they must be of the same order.  We cannot add the matrices in a different order.

4. What is the Additive identity property of the Addition of matrices?

For a matrix A, there is a unique matrix O such that A + O = A = O + A.