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]