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 = [aij]m × n and B = [bij]m × n

Let A + B = C = [cij]m × n and B + A = D = [dij]m × n

Then, cij = aij + bij

= bij + aij , (by using the definition of addition of matrices)

= dij

Since C and D are of the same order and cij = dij 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 = [aij]m × n ,B = [bij]m × n and C = [cij]m × n

Let B + C = D = [dij]m × n , A + B = E = [eij]m × n , A + D = P = [pij]m × n , E + C = Q = [qij]m × n

Then, dij = bij + cij , eij = aij + bij , pij = aij + dij and qij = eij + cij

Now, A + (B + C) = A + D = P = [pij]m × n

and (A + B) + C = E + C = Q = [qij]m × n

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

pij = aij + dij = aij + (bij + cij)

= (aij + bij) + cij , (by the definition of addition of matrices)

= eij + cij

= qij

Since P and Q are of the same order and pij = qij 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 = [aij]m × n and O = [0]m × n

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

= [aij + 0]

= [aij]

= A

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

= [0 + aij]

= [aij]

= 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 = [aij]m × n

Therefore, - A = [- aij]m × n

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

= [aij + (- aij)]

= [0]

= O

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

= [(-aij) + aij]

= [0]

= O

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

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

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Recent Articles

1. Estimating Sum and Difference | Reasonable Estimate | Procedure | Math

May 22, 24 06:21 PM

The procedure of estimating sum and difference are in the following examples. Example 1: Estimate the sum 5290 + 17986 by estimating the numbers to their nearest (i) hundreds (ii) thousands.

2. Round off to Nearest 1000 |Rounding Numbers to Nearest Thousand| Rules

May 22, 24 06:14 PM

While rounding off to the nearest thousand, if the digit in the hundreds place is between 0 – 4 i.e., < 5, then the hundreds place is replaced by ‘0’. If the digit in the hundreds place is = to or > 5…

3. Round off to Nearest 100 | Rounding Numbers To Nearest Hundred | Rules

May 22, 24 05:17 PM

While rounding off to the nearest hundred, if the digit in the tens place is between 0 – 4 i.e. < 5, then the tens place is replaced by ‘0’. If the digit in the units place is equal to or >5, then the…

4. Round off to Nearest 10 |How To Round off to Nearest 10?|Rounding Rule

May 22, 24 03:49 PM

Round off to nearest 10 is discussed here. Rounding can be done for every place-value of number. To round off a number to the nearest tens, we round off to the nearest multiple of ten. A large number…