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.
Addition of Matrices with Example
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.
Conformability for Addition or Subtraction
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
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 of matrices
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] \\
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] \\
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 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]
Associative Property w.r.t Addition
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
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
Answer: Same
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
Answer: Corresponding
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
Answer: Possible
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
Answer: Not possible
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
Answer:
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
Answer: Commutative
8. A+(B+C)=(A+B)+C is ________ law under addition.
O Commutative
O Associative
O Distributive
O None of these
Answer: Associative
9. A+0 \ or \ 0+A = ________
O B
O A
O 0
O All of these
Answer: A
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 Additive identity
O Zero
O Null
O All of these
Answer: All of these
11. The ________ matrix is the additive identity for addition.
O Identity
O Scalar
O Null
O None of these
Answer: Null
Explanation:
When a null matrix is added to any matrix, the result is that matrix which shows the additive identity.
12. The null matrix is the additive ________ for addition.
O Identity
O Scalar
O Null
O None of these
Answer: Identity
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:
Zero is the additive identity.
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
Answer: Zero
16. If A+B=0 then B is the _______ inverse of A
O Multiplicative
O Additive
O Zero
O None of these
Answer: Additive
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 Additive
O Multiplicative
O Identity
O All of these
Answer: Additive
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]
O Row
O Column
O Same
O Different
Answer: Same
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
Answer: Corresponding
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
Answer: Possible
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
Answer: Not possible
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
Answer:
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
Answer: Commutative
8. A+(B+C)=(A+B)+C is ________ law under addition.
O Commutative
O Associative
O Distributive
O None of these
Answer: Associative
9. A+0 \ or \ 0+A = ________
O B
O A
O 0
O All of these
Answer: A
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 Additive identity
O Zero
O Null
O All of these
Answer: All of these
11. The ________ matrix is the additive identity for addition.
O Identity
O Scalar
O Null
O None of these
Answer: Null
Explanation:
When a null matrix is added to any matrix, the result is that matrix which shows the additive identity.
12. The null matrix is the additive ________ for addition.
O Identity
O Scalar
O Null
O None of these
Answer: Identity
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:
Zero is the additive identity.
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
Answer: Zero
16. If A+B=0 then B is the _______ inverse of A
O Multiplicative
O Additive
O Zero
O None of these
Answer: Additive
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 Additive
O Multiplicative
O Identity
O All of these
Answer: Additive
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]