Question

Properties of Matrices Addition

Answer 1

1. Commutative Property: If A and B are two matrices of the same order, then their sum is commutative, i.e.,

A + B = B + A

Proof: Let A = [a_ij - b_ij]_{m × n} and B = [b_ij - b_ij]_{m × n}

So, two matrices are suitable for addition, then

A + B = [a_ij - b_ij]_{m × n} + [b_ij]_{m × n}

= [a_ij + b_ij]_{m × n}

By definition of addition of matrices

= [b_ij + a_ij]_{m × n}

(∵ a_ij and b_ij are number. Thus their addition is communtative)

= [b_ij]_{m × n} + [a_ij]_{m × n}

(By definition of addition of matrices)

= B + A

Thus, A + B - B + A

1. e., Sum of two matrices is commutative.

2. Associative Property: If A B and C are three matrices of order m × n, then their sum is associative, i.e.,

(A + B) + C = A + (B + C)

Proof : Let A = [a_ij]_{m × n}; B = [b_ij]_{m × n}; C = [c_ij]_{m × n}

(A + B) + C =([a_ij]_{m × n} + [b_ij]_{m × n}) + [c_ij]_{m × n}

= [a_ij + b_ij]_{m × n} + [c_ij]_{m × n} (By definition of addition)

= [(a_ij + b_ij) + [c_ij]_{m × n} (By definition of addition)

= [a_ij + (b_ij + [c_ij)]_{m × n} (∵ a_ij, b_ij, [c_ij, are numbers whose sum is commutative)

= [a_ij]_{m × n} + [b_ij + c_ij]_{m × n} (By definition of addition of two matrices)

= [a_ij]_{m × n} + [b_ij]_{m × n} + [c_ij]_{m × n}

= A + (B + C)

(A + B) + C = A + (B + C)

Thus, sum of three matrices is associative.

3. Existence of Additive Identity : If order of matrix A is m × n and also the order of matrix O is m × n, then

A + O = A = O + A

It means matrix O_{m × n} is additive identity.

Proof:

Let A = [a_ij]_{m × n}

A + O = [a_ij]_{m × n} + [O_ij]_{m × n}

= [a_ij + O]_{m × n} = [a_ij]_{m × n} = A

and O + A = [O + ai_ij]_{m × n} = [a_ij]_{m × n} = A

4. Existence of Additive Inverse : For each matrix A there exist matrix - A of same order such that A + (- A) = O, where O is null matrix.

Then - A is called additive inverse of A or negative of A.

Proof : Let A = [a_ij]_{m × n} then - A = - [a_ij]_{m × n}

Thus, A + (- A) = [a_ij]_{m × n} + [- a_ij]_{m × n}

= [a_ij + (-a_ij]_{m × n} (By definition of addition of matrices)

= [a_ij - a_ij]_{m × n} = [0]_{m × n} = O_{m × n}

5. Cancellation Laws : If A, B, C are matrices of the same order, then A + B = A + C ⇒ B = C,

Proof :

We have A + B = A + C ...(i)

Adding - A to both sides, we get

- A + (A + B) = - A + (A + C)

By associativity, we get

(- A + A) + B = (- A + A) + C

⇒ O + B = O + C

[Since, O is the additive identity]

B = C [∵ O + B = B, O + C = C By additive identity definition]