Simple Properties of Addition

A matrix is a rectangular array of numbers called elements or entries into rows and columns. The dimension of the matrix gives the number of rows and columns of the matrix in that order. For adding the two matrices of the same dimensions, just add the entries of the matrix in the corresponding position.

Addition of Vectors 

For drawing the vectors geometrically, draw the vectors to a common scale and then place them according to head. The vector that connects the tail of the first vector to the head of the second vector is the sum. 

Points to be kept in mind while adding two vectors:

• Vectors behave independently of each other.
• Vector addition is finding the resultant of a number of vectors acting upon a body.
• Commutative law is applied in vector addition which means the resultant vector is independent of the order of vectors.
• Only matrices with the same order can be added to each other.
• Vectors can be added geometrically only.

Dimension Considerations

The sum of two matrices is always another matrix. This describes the closure property of matrix addition.

When adding two matrices, make sure the matrices have the same dimensions. In order of words, we can add a 2 x 3 with a 2 x 3 or a 2 x 2 with a 2 x 2. However, for adding a 3 x 2 with a 2 x 3 or a 2 x 2 with a 3 x 3, addition is not possible yet.

If the dimensions of two matrices are different, the addition is not defined because some of the entries in the matrix will not have the corresponding entry in the matrix.

Matrix Addition and Real Numbers Addition

As matrix addition relies much on the addition of the real numbers, many of the additional properties that are known to us and are true with real numbers are also found true with matrices.

Properties of Matrix Addition

The algebraic operations of addition, subtraction, multiplication, and inverse multiplication of matrices and operations involving multiple types of matrices can be done efficiently using the characteristics of matrices. The study of matrix property also includes additive, multiplicative identity, and inverse matrices.

1. Commutative property of addition:

This property states that we can add two matrices in any order and get the same result. This parallels the commutative property of addition for real numbers. 

2. Associative property of addition:

This property states that we can change the groups in matrix addition and get the same result.

In each column, we simplify one side of the identity into a single matrix. The two resulting matrices are equivalent to the real number associative property of addition. We do not need any guidelines that tell which addition is to be performed first as it hardly matters.

3. Additive identity property:

A zero matrix is a matrix in which all of the entries are 0. When a zero matrix is added to any matrix, the result is always the matrix itself.

It illustrates the additive identity property; that the sum of any matrix and the appropriate zero matrices is the matrix. For all real numbers. the number is the additive identity in the real number system same as the additive identity for matrices.

4. Additive inverse property:

The opposite of a matrix is a matrix, where each element in this matrix is the opposite of the corresponding element in the matrix.

If we add these two matrices,  we get a zero matrix, which illustrates the additive inverse property. The sum of a real number and its opposite is always 0, so the sum of any matrix and its opposite gives a zero matrix. Because of this, we refer to opposite matrices as additive inverse.

5. Closure property of addition

The closure property of addition can be illustrated with the help of real numbers. Since the set of real numbers is closed under addition, we will get another real number when we add two real numbers. Here, there will be no possibility of getting a complex number other than real numbers.

Mathematical Presentation of Properties

 1. Commutative law

If A = [aij], B = [bij] are two matrix of the same order  m × n.

So, A + B = B + A.

 2. Associative law

Any three matrices A = [aij], B = [bij] & C = [cij] with the same order  m × n.

Then (A + B) + C = A + (B + C).

3. Additive identity property

Let A = [aij] be an m × n matrix and O be an m × n zero matrix.

Then A + O = O + A = A where O is the additive identity for matrix addition.

4. Additive inverse property

Let A = [aij]m×n be any matrix, then we have another matrix as – A = [–aij]m×n

 Then A + (–A) = (–A) + A= O where – A is the additive inverse of A or negative element of A.

5. Closure of property 

For any three matrices, A = [aij], B = [bij], C = [cij] of the same order, say m × n.

A + B = C, where C is a matrix with the same dimensions as A and B.

Conclusion

We have discussed the definition of the matrix, the transpose, and the inverse of the matrix. We studied the properties of a matrix addition such as cumulative, associative, identity, inverse, and closure laws and their mathematical presentation. We have also discussed the properties of the transpose and inverse matrix.