Properties of Addition of Matrices

We will discuss about the properties of addition of matrices.

1. Commutative Law of Addition of Matrix: Matrix multiplication is commutative. This says that, if A and B are matrices of the same order such that A + B is defined then A + B = B + A.

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

Let A + B = C = [c_ij]_{m × n} and B + A = D = [d_ij]_{m × n}

Then, c_ij = a_ij + b_ij

              = b_ij + a_ij , (by using the definition of addition of matrices)

              = d_ij

Since C and D are of the same order and c_ij = d_ij then, C = D.

i.e., A + B = B + A. This completes the proof.

2. Associative Law of Addition of Matrix: Matrix addition is associative. This says that, if A, B and C are Three matrices of the same order such that the matrices B + C, A + (B + C), A + B, (A + B) + C are defined then A + (B + C) = (A + B) + C.

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

Let B + C = D = [d_ij]_{m × n} , A + B = E = [e_ij]_{m × n} , A + D = P = [p_ij]_{m × n} , E + C = Q = [q_ij]_{m × n}

Then, d_ij = b_ij + c_ij , e_ij = a_ij + b_ij , p_ij = a_ij + d_ij and q_ij = e_ij + c_ij

Now, A + (B + C) = A + D = P = [p_ij]_{m × n}

and (A + B) + C = E + C = Q = [q_ij]_{m × n}

Therefore, P and Q are the matrices of the same order and

              p_ij = a_ij + d_ij = a_ij + (b_ij + c_ij)

                   = (a_ij + b_ij) + c_ij , (by the definition of addition of matrices)

                    = e_ij + c_ij

                    = q_ij

Since P and Q are of the same order and p_ij = q_ij then, P = Q.

i.e., A + (B + C) = (A + B) + C. This completes the proof.

3. Existence of Additive Identity of Matrix: Let A be the matrix then, A + O = A = O + A

Therefore, ‘O’ is the null matrix of the same order as the matrix A

Proof: Let A = [a_ij]_{m × n} and O = [0]_{m × n}

Therefore, A + O = [a_ij] + [0]

                          = [a_ij + 0]

                          = [a_ij]

                           = A

Again, O + A = [0] + [a_ij]

                     = [0 + a_ij]

                     = [a_ij]

                     = A

Note: The null matrix is called the additive identity for the matrices.

4. Existence of Additive Inverse of Matrix: Let A be the matrix then, A + (- A) = O = (- A) + A

Proof: Let A = [a_ij]_{m × n}

Therefore, - A = [- a_ij]_{m × n}

Now, A + (- A) = [a_ij] + [- a_ij]

                       = [a_ij + (- a_ij)]

                       = [0]

                        = O

Again (- A) + A = [- a_ij] + [a_ij]

                       = [(-a_ij) + a_ij]

                       = [0]

                       = O

Therefore, A + (- A) = O = (- A) + A

Note: The matrix – A is called the additive inverse of the matrix A.

10th Grade Math

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