Addition of Matrices Worksheet

Matrices are a fundamental concept in mathematics, particularly in linear algebra. They are rectangular arrays of numbers arranged in rows and columns.

This worksheet will help students grasp the concept of matrix addition, improve their problem-solving skills, and prepare them for more advanced topics in mathematics.

1. Given matrices A and B, where

A = \begin{bmatrix} 1 & 3\\ 5 & 7 \end{bmatrix} and B = \begin{bmatrix} 2 & 4\\ 6 & 8 \end{bmatrix}

Find the matrix C = A + B

To find the sum of matrix A and B is C

c₁₁ = a₁₁ + b₁₁ = 1 + 2 = 3

c₁₂ = a₁₂ + b₁₂ = 3 + 4 = 7

c₂₁ = a₂₁ + b₂₁ = 5 + 6 = 11

c₂₂ = a₂₂ +b₂₂ = 7 + 8 = 15

Resulting matrix C is

C = \begin{bmatrix} 3 & 7\\ 11 & 15 \end{bmatrix}

2. Let A = \begin{bmatrix} 0 & -1\\ 3 & 2 \end{bmatrix} and B = \begin{bmatrix} 5 & 3\\ -3 & 4 \end{bmatrix}

Compute the elements of matrix C = A + B.

To find the elements of matrix C

c₁₁ = a₁₁ + b₁₁ = 0 + 5 = 5

c₁₂ = a₁₂ + b₁₂ = -1 + 3 = 2

c₂₁ = a₂₁ + b₂₁ = 3 - 3 = 0

c₂₂ = a₂₂ +b₂₂ = 2 + 4 = 6

Resulting matrix C is

C = \begin{bmatrix} 5 & 2\\ 0 & 6 \end{bmatrix}

3. Verify the commutative property for matrices

A = \begin{bmatrix} 1 & 3\\ 2 & 4 \end{bmatrix}( 1 3 ) and B = \begin{bmatrix} 4 & 3\\ 2 & 1 \end{bmatrix}

To verify the commutative property for matrices follow these steps

First we calculate A + B

c₁₁ = a₁₁ + b₁₁ = 1 + 4 = 5

c₁₂ = a₁₂ + b₁₂ = 3 + 3 = 6

c₂₁ = a₂₁ + b₂₁ = 2 + 2 = 4

c₂₂ = a₂₂ +b₂₂ = 4 + 1 = 5

Resulting matrix C₁ is

C₁ = \begin{bmatrix} 5 & 6\\ 4 & 5 \end{bmatrix}

Now, we calculate B + A

c₁₁ = a₁₁ + b₁₁ = 4 + 1 = 5

c₁₂ = a₁₂ + b₁₂ = 3 + 3 = 6

c₂₁ = a₂₁ + b₂₁ = 2 + 2 = 4

c₂₂ = a₂₂ +b₂₂ = 1 + 4 = 5

The resulting matrix C is

C₂ = \begin{bmatrix} 5 & 6\\ 4 & 5 \end{bmatrix}

So, C₁ and C₂ are equal so, it follow the commutative property.

4. Given Matrices are

A = \begin{bmatrix} 1 & 0\\ 4 & 5 \end{bmatrix}and B = \begin{bmatrix} 3 & 1\\ -2 & 6 \end{bmatrix}

Show that A + B = B + A.

First we calculate A + B

c₁₁ = a₁₁ + b₁₁ = -1 + 3 = 2

c₁₂ = a₁₂ + b₁₂ = 0 + 1 = 1

c₂₁ = a₂₁ + b₂₁ = 4 - 2 = 2

c₂₂ = a₂₂ +b₂₂ = 5 + 6 = 11

Resulting matrix C₁ is

C₁ = \begin{bmatrix} 2 & 1\\ 2 & 11 \end{bmatrix}

Now, we calculate B + A

c₁₁ = a₁₁ + b₁₁ = 3 + (-1) = 2

c₁₂ = a₁₂ + b₁₂ = 1 + 0 = 1

c₂₁ = a₂₁ + b₂₁ = - 2 + 4 = 2

c₂₂ = a₂₂ +b₂₂ = 6 + 5 = 11

Resulting matrix C₂ is

C₂ = \begin{bmatrix} 2 & 1\\ 2 & 11 \end{bmatrix}

So, C₁ and C₂ are equal. Therefore A + B = B + A.

5. Given Matrices are

A = \begin{bmatrix} 2 & 1\\ 0 & -1 \end{bmatrix}, B = \begin{bmatrix} 3 & 5\\ 6 & 7 \end{bmatrix} and C = \begin{bmatrix} 4 & 8\\ 2 & 3 \end{bmatrix}

Verify the associative property (A + B) + C = A + (B + C).

First we calculate A + B

e₁₁ = a₁₁ + b₁₁ = 2 + 3 = 5

e₁₂ = a₁₂ + b₁₂ = 1 + 5 = 6

e₂₁ = a₂₁ + b₂₁ = 0 + 6 = 6

e₂₂ = a₂₂ + b₂₂ = -1 + 7 = 6

Resulting matrix E₁ is

E₁ = \begin{bmatrix} 5 & 6\\ 6 & 6 \end{bmatrix}

(A + B) + C

d₁₁ = c₁₁ + e₁₁ = 4 + 5 = 9

d₁₂ = c₁₂ + e₁₂ = 8 + 6 = 14

d₂₁ = c₂₁ + e₂₁ = 2 + 6 = 8

d₂₂ = c₂₂ +e₂₂ = 3 + 6 = 9

So, (A + B) + C = \begin{bmatrix} 9 & 14\\ 8 & 9 \end{bmatrix}

Now, we calculate B + C

e₁₁ = c₁₁ + b₁₁ = 4 + 3 = 7

e₁₂ = c₁₂ + b₁₂ = 8 + 5 = 13

e₂₁ = c₂₁ + b₂₁ = 2 + 6 = 8

e₂₂ = c₂₂ +b₂₂ = 3 + 7 = 10

Resulting matrix E₂ is

E₂ = \begin{bmatrix} 7 & 13\\ 8 & 10 \end{bmatrix}

A + (B + C)

d₁₁ = a₁₁ + e₁₁ = 2 + 7 = 9

d₁₂ = a₁₂ + e₁₂ = 1 + 13 = 14

d₂₁ = a₂₁ + e₂₁ = 0 + 8 = 8

d₂₂ = a₂₂ + e₂₂ = -1 + 10 = 9

So, A + (B + C) = \begin{bmatrix} 9 & 14\\ 8 & 9 \end{bmatrix}

So, (A + B) + C = A + (B + C).

6. If A = \begin{bmatrix} 1 & 4\\ 5 & 7 \end{bmatrix}, B = \begin{bmatrix} 0 & 2\\ 3 & 6 \end{bmatrix} and C = \begin{bmatrix} 2 & 1\\ 4 & 5 \end{bmatrix}

Prove that matrix addition is associative by calculating both (A + B) + C and A + (B + C).

First we calculate A + B

e₁₁ = a₁₁ + b₁₁ = 1 + 0 = 1

e₁₂ = a₁₂ + b₁₂ = 4 + 2 = 6

e₂₁ = a₂₁ + b₂₁ = 5 + 3 = 8

e₂₂ = a₂₂ + b₂₂ = 7 + 6 = 13

Resulting matrix E₁ is

E₁ = \begin{bmatrix} 1 & 6\\ 8 & 13 \end{bmatrix}

(A + B) + C

d₁₁ = c₁₁ + e₁₁ = 2 + 1 = 3

d₁₂ = c₁₂ + e₁₂ = 1 + 6 = 7

d₂₁ = c₂₁ + e₂₁ = 4 + 8 = 12

d₂₂ = c₂₂ +e₂₂ = 5 + 13 = 18

So, (A + B) + C = \begin{bmatrix} 3 & 7\\ 12 & 8 \end{bmatrix}

Now, we calculate B + C

e₁₁ = c₁₁ + b₁₁ = 2 + 0 = 2

e₁₂ = c₁₂ + b₁₂ = 1 + 2 = 3

e₂₁ = c₂₁ + b₂₁ = 4 + 3 = 7

e₂₂ = c₂₂ +b₂₂ = 5 + 6 = 11

Resulting matrix E₂ is

E₂ = \begin{bmatrix} 2 & 3\\ 7 & 11 \end{bmatrix}

A + (B + C)

d₁₁ = a₁₁ + e₁₁ = 1 + 2 = 3

d₁₂ = a₁₂ + e₁₂ = 4 + 3 = 7

d₂₁ = a₂₁ + e₂₁ = 5 + 7 = 12

d₂₂ = a₂₂ +e₂₂ = 7 + 11 = 18

So, A + (B + C) = \begin{bmatrix} 3 & 7\\ 12 & 8 \end{bmatrix}

So, (A + B) + C = A + (B + C).

7. Given matrix A = \begin{bmatrix} 6 & 7\\ 8 & 9 \end{bmatrix} and the zero matrix O = \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}, show that A +O = A.

A + O = \begin{bmatrix} 6 & 7\\ 8 & 9 \end{bmatrix} + \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}

c₁₁ = a₁₁ + 0₁₁ = 6 + 0 = 6

c₁₂ = a₁₂ + 0₁₂ = 8 + 0 = 8

c₂₁ = a₂₁ + 0₂₁ = 7 + 0 = 7

c₂₂ = a₂₂ +0₂₂ = 9 + 0 = 9

So, Addition of matrix A and O is \begin{bmatrix} 6 & 8\\ 7 & 9 \end{bmatrix}

It is equal to A matrix.

8. Given matrix A = \begin{bmatrix} -3 & 5\\ 7 & 2 \end{bmatrix}, find the result of A + zero matrix.

A + O = \begin{bmatrix} -3 & 5\\ 7 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}

c₁₁ = a₁₁ + O₁₁ = -3 + 0 = -3

c₁₂ = a₁₂ + O₁₂ = 5 + 0 = 5

c₂₁ = a₂₁ + O₂₁ = 7 + 0 = 7

c₂₂ = a₂₂ +O₂₂ = 2 + 0 = 2

So, Addition of matrix A and O is \begin{bmatrix} -3 & 5\\ 7 & 2 \end{bmatrix}

9. Given matrix A = \begin{bmatrix} 2 & -3\\ 4 & -5 \end{bmatrix}, find the additive inverse −A and verify that A + (−A) = O.

Additive inverse -A = \begin{bmatrix} -2 & 3\\ -4 & 5 \end{bmatrix}

So, A + (-A) = \begin{bmatrix} 2 & -3\\ 4 & -5 \end{bmatrix} + \begin{bmatrix} -2 & 3\\ -4 & 5 \end{bmatrix}

= \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}

So, A + (-A) = O.

1O. Given matrix A = \begin{bmatrix} 3 & 4\\ -2 & 1 \end{bmatrix}, find the additive inverse −A and verify that A + (−A) = O.

Additive inverse -A = \begin{bmatrix} 3 & 4\\ -2 & 1 \end{bmatrix}

So, A + (-A) = \begin{bmatrix} 3 & 4\\ -2 & 1 \end{bmatrix} + \begin{bmatrix} -3 & -4\\ 2 & -1 \end{bmatrix}

= \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}

So, A + (-A) = O.

Practice Questions