In this blog post, you will find how to add and subtract matrices step by step with examples and MCQs to help you to understand the concept. Whether you are new to matrices or need a refresher, this post is perfect.
After covering this, you will be able to learn about the basic concepts related to Addition, Subtraction of matrices and other related ideas related to Matrix Addition.

When two matrices have the same order, they are conformable for Addition and Subtraction. Two matrices have the same order means “the same number of Rows and Columns”.

Addition can be obtained by adding the corresponding elements of the matrices.

When two matrices have the same order, they are conformable for Addition and Subtraction.
Two matrices have the same order means “the same number of Rows and Columns”
A=\left[\begin{array}{ll}3 & 8 \\4 & 6\end{array}\right], B=\left[\begin{array}{cc}4 & 0 \\1 & -9\end{array}\right] \\
C=\left[\begin{array}{ccc} -3 & 4 & -5 \\ 2 & 3 & 1 \end{array}\right]

D=\left[\begin{array}{ccc} -3 & -4 & 5 \\ 1 & 2 & 3 \end{array}\right] \\
Q: Which of the following matrix operation are defined?
1. A+B
As the order of A is 2-by-2
And the order of B is 2-by-2
Hence orders are same
So A+B are conformable
2. B+D
As the order of B is 2-by-2
And the order of D is 2-by-3
So B+D are not conformable
3. A-C
As the order of A is 2-by-2
And the order of C is 2-by-3
So A-C are not conformable
4. C-D
As the order of C is 2-by-3
And the order of D is 2-by-3
So C-D are conformable
Addition can be obtained by adding the corresponding elements of the matrices.
Example:
A=\left[\begin{array}{ll}3 & 8 \\4 & 6\end{array}\right], B=\left[\begin{array}{cc}4 & 0 \\1 & -9\end{array}\right] \\
A+B=\left[\begin{array}{ll}3 & 8 \\4 & 6\end{array}\right]+\left[\begin{array}{cc}4 & 0 \\1 -9\end{array}\right] \\
A+B=\left[\begin{array}{ll}3+4 & 8+0 \\4+1 & 6-9\end{array}\right] \\
A+B=\left[\begin{array}{cc}7 & 8 \\5 & -3\end{array}\right] \\
Subtraction of matrices
Subtraction can be obtained by subtracting the corresponding elements of the matrices.
Example:
A=\left[\begin{array}{ll}3 & 8 \\4 & 6\end{array}\right], B=\left[\begin{array}{cc}4 & 0 \\1 & -9\end{array}\right] \\
A-B
=\left[\begin{array}{ll}3 & 8 \\4 & 6\end{array}\right]-\left[\begin{array}{cc}4 & 0 \\1 -9\end{array}\right] \\
A-B=\left[\begin{array}{ll}3-4 & 8-0 \\4-1 & 6+9\end{array}\right] \\
A-B=\left[\begin{array}{cc}-1 & 8 \\3 & 15\end{array}\right] \\
Commutative Property of matrix w.r.t Addition
If two matrices of same order then A+B=B+A is called the Commutative law under addition.
C=\left[\begin{array}{ccc} -3 & 4 & -5 \\ 2 & 3 & 1 \end{array}\right]

D=\left[\begin{array}{ccc} -3 & -4 & 5 \\ 1 & 2 & 3 \end{array}\right] \\
{C}+{D}={D}+ {C} \\
LHS:
C+D
=\left[\begin{array}{ccc} -3 & 4 & -5 \\ 2 & 3 & 1 \end{array}\right]+\left[\begin{array}{ccc} -3 & -4 & 5 \\ 1 & 2 & 3 \end{array}\right] \\
=\left[\begin{array}{ccc} -3-3 & 4-4 & -5+5 \\ 2+1 & 3+2 & 1+3 \end{array}\right] \\
=\left[\begin{array}{ccc} -6 & 0 & 0 \\ 3 & 5 & 4 \end{array}\right]
RHS
D+C
=\left[\begin{array}{ccc}-3 & -4 & 5 \\ 1 & 2 & 3\end{array}\right]+\left[\begin{array}{ccc}-3 & 4 & -5 \\ 2 & 3 & 1\end{array}\right]
=\left[\begin{array}{ccc}-3-3 & -4+4 & 5-5 \\ 1+2 & 2+3 & 3+1\end{array}\right]
=\left[\begin{array}{ccc}-6 & 0 & 0 \\ 3 & 5 & 4\end{array}\right]
If three matrices of same order, then
A+(B+C)=(A+B)+C is called the Associative law under addition.
Verify
{A}+({B}+{C})=({A}+{B})+{C}
for the following matrices.
A=\left[\begin{array}{cc}2 & -3 \\ 4 & 1\end{array}\right],
B=\left[\begin{array}{cc}5 & -2 \\ 3 & 6\end{array}\right],
C=\left[\begin{array}{cc}1 & 7 \\ -6 & -3\end{array}\right]
Solution:
{A}+({B}+{C})=({A}+{B})+{C}
LHS: A+(B+C)
B+C
=\left[\begin{array}{cc}5 & -2 \\ 3 & 6\end{array}\right]+\left[\begin{array}{cc}1 & 7 \\ -6 & -3\end{array}\right]
=\left[\begin{array}{cc}5+1 & -2+7 \\ 3-6 & 6-3\end{array}\right]
=\left[\begin{array}{cc}6 & 5 \\ -3 & 3\end{array}\right]
A+(B+C)
=\left[\begin{array}{cc}2 & -3 \\ 4 & 1\end{array}\right]+\left[\begin{array}{cc}6 & 5 \\ -3 & 3\end{array}\right]
=\left[\begin{array}{cc}2+6 & -3+5 \\ 4-3 & 1+3\end{array}\right]
=\left[\begin{array}{ll}8 & 2 \\ 1 & 4\end{array}\right]
RHS: (A+B)+C
A+B
=\left[\begin{array}{cc}2 & -3 \\ 4 & 1\end{array}\right]+\left[\begin{array}{cc}5 & -2 \\ 3 & 6\end{array}\right]
=\left[\begin{array}{cc}2+5 & -3-2 \\ 4+3 & 1+6\end{array}\right]
=\left[\begin{array}{cc}7 & -5 \\ 7 & 7\end{array}\right]
(A+B)+C
=\left[\begin{array}{cc}7 & -5 \\ 7 & 7\end{array}\right]+\left[\begin{array}{cc}1 & 7 \\ -6 & -3\end{array}\right]
=\left[\begin{array}{cc}7+1 & -5+7 \\ 7-6 & 7-3\end{array}\right]
=\left[\begin{array}{ll}8 & 2 \\ 1 & 4\end{array}\right]
Hence
A+(B+C)=(A+B)+C
Proved

Addition of Matrices MCQs with Explanation

1. Two matrices are conformable for addition/subtraction, if they are of the ________ order.
O Row
O Column
O Same
O Different

Explanation:
When the number of rows and columns of both the matrices are same, then both the matrices can be added/subtracted.

2. Addition of two matrices is obtained by adding the ________ elements of the matrices.
O Row
O Column
O Corresponding
O Different

Explanation:
For Addition of matrices, it must be noted to add the corresponding elements.

3. The addition of A=\left[\begin{array}{lll}1 & -2 & 3\end{array}\right] \ and \ B=\left[\begin{array}{lll}a & b & c\end{array}\right] is ________.
O Possible
O Not possible
O Identity matrix
O None of these

Explanation:
The addition of Matrix A and B is possible, because the order of both the matrices are same, that is:
1-by-3

4. The addition of A=\left[\begin{array}{lll}1 & 2 \end{array}\right] \ and \ B=\left[\begin{array}{lll}a & b & c\end{array}\right] is ________
O Possible
O Not possible
O Identity matrix
O None of these

Explanation:
The addition of Matrix A and B is not possible, because the order of both the matrices are not same.

5. The real number is multiplying to ________ elements of the matrix.
O Each
O Row
O Column
O All of these

Explanation:
The real number is multiplied to each elements of the matrix. Example is in MCQs No. 6

6. If A=\left[\begin{array}{lll}1 & -2 & 3\end{array}\right] \ then \ -3A= ________
O \left[\begin{array}{lll}-3 & -2 & -9\end{array}\right]
O \left[\begin{array}{lll}-1 & 2 & 3\end{array}\right]
O \left[\begin{array}{lll}-3 & 6 & -9\end{array}\right]
O None of the above

Answer: \left[\begin{array}{lll}-3 & 6 & -9\end{array}\right]
Explanation:
A=\left[\begin{array}{lll}1 & -2 & 3\end{array}\right]
-3A=-3\left[\begin{array}{lll}1 & -2 & 3\end{array}\right]
-3A=\left[\begin{array}{lll}-3 & 6 & -9\end{array}\right]

7. A+B=B+A is ________ property under addition.
O Commutative
O Associative
O Distributive
O None of these

8. A+(B+C)=(A+B)+C is ________ law under addition.
O Commutative
O Associative
O Distributive
O None of these

9. A+0 \ or \ 0+A = ________
O B
O A
O 0
O All of these

Explanation:
When zero (0) is added to any number, the answer should be that mumber.
0+3=3

10. \left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right] is ________ matrix.
O Zero
O Null
O All of these

O Identity
O Scalar
O Null
O None of these

Explanation:
When a null matrix is added to any matrix, the result is that matrix which shows the additive identity.

O Identity
O Scalar
O Null
O None of these

Explanation:
When a null matrix is added to any matrix, the result is that matrix which shows the additive identity.

13. The additive identity of \left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right] is ________
O \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
O \left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]
O \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]
O \left[\begin{array}{ll}-1 & -3 \\ -2 & -4\end{array}\right]

Answer: \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
Explanation:

14. The additive identity of \left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right] is ________
O {\left[\begin{array}{ll}0 \end{array}\right] }
O {\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] }
O {\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right] }
O \left[\begin{array}{ll}-1 & -3 \\ -2 & -4\end{array}\right]

Answer: {\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] }
Explanation:
The order of null matrix is same of that matrix.

15. In additive inverse, the sum of two matrices are ________
O Identity
O Zero
O a & b
O None of these

16. If A+B=0 then B is the _______ inverse of A
O Multiplicative
O Zero
O None of these

Explanation:
When the sum of two numbers is zero, then they are the additive inverse of each other.

17. If P+Q=0 , then P \ and \ Q are the ________ inverse of each other.
O Multiplicative
O Identity
O All of these

Explanation:
When the sum of two numbers is zero, then they are the additive inverse of each other.

18. The additive inverse of \left[\begin{array}{ll}1 & -3 \\ 2 & 4\end{array}\right] is ________
O \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
O \left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]
O \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]
O \left[\begin{array}{ll}-1 & 3 \\ -2 & -4\end{array}\right]

Answer: \left[\begin{array}{ll}-1 & 3 \\ -2 & -4\end{array}\right]
Explanation:
when the sum of two matrices is zero OR the matrices of opposite signs.
\left[\begin{array}{ll}1 & -3 \\ 2 & 4\end{array}\right]+\left[\begin{array}{ll}-1 & 3 \\ -2 & -4\end{array}\right] =\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]