[Maths Class Notes] on Matrix Addition Pdf for Exam

Before discussing the operations of the matrix, let’s discuss what a matrix is.

A matrix is a rectangular (2D) array of numbers or symbols which are generally arranged in rows and columns. One can think of it as a table of numbers.
The order of the matrix is defined as the number of rows and columns.
The entries are the numbers in the matrix and each number is known as an element.
The plural of matrix is matrices.
The size of a matrix is referred to as ‘n by m’ matrix and is written as m×n, where n is the number of rows and m is the number of columns.
For example, we have a 3×2 matrix, that’s because the number of rows here is equal to 3 and the number of columns is equal to 2.

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The dimensions of a matrix can be defined as the number of rows and columns of the matrix in that order. Since matrix A given above has 2 rows and 3 columns, it is known as a 2×3 matrix.

What are the Different Types of Matrix?

There are different types of matrices. Here they are –

1) Row matrix

2) Column matrix

3) Null matrix

4) Square matrix

5) Diagonal matrix

6) Upper triangular matrix

7) Lower triangular matrix

8) Symmetric matrix

9) Anti-symmetric matrix

Adding Matrices

Two matrices must have an equal number of columns and rows in order to be added. The sum of any two matrices suppose A and B will be a matrix which has the same number of rows and columns as do the matrices A and B. The sum of A and B, can be denoted as A + B, is computed by adding corresponding elements of A and B.

A + B = [begin{bmatrix} a_{11} & a_{12} & cdots & a_{1n}\ a_{21} & a_{22} & cdots & a_{2n}\ vdots & vdots & ddots & vdots\ a_{m1} & a_{m2} & cdots & a_{mn} end {bmatrix}] + [begin{bmatrix} b_{11} & b_{12} & cdots & b_{1n}\ b_{21} & b_{22} & cdots & b_{2n}\ vdots & vdots & ddots & vdots\ b_{m1} & b_{m2} & cdots & b_{mn} end {bmatrix}]

= [begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & cdots & a_{1n} + b_{1n}\ a_{21} +b_{21} & a_{22} + b_{21}& cdots & a_{2n} + b_{2n}\ vdots & vdots & ddots & vdots\ a_{m1} + b_{m1} & a_{m2} + b_{m2} & cdots & a_{mn} + b_{mn} end {bmatrix}]

Matrix Sums and Answers

Let us suppose that we have two matrices A and B.

Both the matrices A and B have the same number of rows and columns (that is the number of rows is 2 and the number of columns is 3), so they can be added. In order words, you can add a 2 x 3 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 2 x 2 matrix. However, you cannot add a 3 x 2 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 3 x 3 matrix.

A = [begin{bmatrix} 1 & 2 & 3\ 7 & 8 & 9end {bmatrix}] B = [begin{bmatrix} 5 & 6 & 7\ 3 & 4 & 5end {bmatrix}]

A + B = [begin{bmatrix} 1 + 5 & 2 + 6 & 3+ 7\ 7 + 3& 8 + 4 & 9 + 5end {bmatrix}]

A + B = [begin{bmatrix} 6 & 8 & 10\ 10 & 12 & 14end {bmatrix}]

Note: Keep in mind that the order in which matrices are added is not important; thus, we can say that A + B = B + A.

Properties of Matrix Addition

1. The Commutative Law	If matrix A = [aij] and matrix B = [bij] are the matrices of the same order, we can say m × n, then A + B will be equal to B + A.
2. The Associative Law	For any three matrices namely A , B, and C, A = [aij], B = [bij], and C = [cij] of the same order, say suppose m × n, then we can write (A + B) + C is equal to A + (B + C).
3. The Existence of Additive Identity	Let us say we have a matrix A = [aij] be an m × n matrix and O be an m × n zero matrix, then A + O is equal to O + A = A. In simpler words, we can say that O is the additive identity for matrix addition.
4. The Existence of Additive Inverse	Let matrix A = [aij]m×n be any matrix, then we have another matrix as – A = [–aij]m×n such that A + (–A) is equal to (–A) + A= O. So, – A can be known as the additive inverse of A or negative of A.

Applications of Matrices

Matrix is used in many branches of mathematics, for example, for calculations related to vectors like finding the derivative, integration, the integral of a matrix, etc.
Matrices are widely used in matrix and linear algebra, in particular, to represent and solve linear systems of equations.
A matrix is also used in solving eigenvalue problems, symmetric and eigenvectors, linear regression, optimization problems, etc.

Solved Example Problems

Question 1. Add the following matrices:

[A = begin{bmatrix}1 & 2&3 \7 & 8&9 end{bmatrix}] [B =begin{bmatrix}5 & 6&7 \3& 4&5 end{bmatrix} ]

Solution: We have two matrices A and B.

Both the matrices A and B have the same number of rows and columns (that is the number of rows is 2 and the number of columns is 3), so they can be added. In other words, you can add a 2 x 3 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 2 x 2 matrix. However, you cannot add a 3 x 2 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 3 x 3 matrix.

[A = begin{bmatrix}1 & 2&3 \7 & 8&9 end{bmatrix}] [B =begin{bmatrix}5 & 6&7 \3& 4&5 end{bmatrix} ]

[ A + B =begin{bmatrix}1+5 & 2+6&3+7 \7+3& 8+4&9+5 end{bmatrix} ]

[ A+B = begin{bmatrix}6 & 8&10 \10 & 12&14 end{bmatrix} ]

Question 2. Add the following matrices.

[ A = begin{bmatrix}3 & 4&9 \12& 11&35 end{bmatrix}] [ B = begin{bmatrix}6 & 2 \5 & 8 end{bmatrix} ]

Solution: Let’s add the following two matrices A and B. As we know that matrices are added entry-wise, we have to a
dd the 3 and the 6, the 12 and 5, the 4 and the 6, and the 11 and the 8. But what do I add to the entries 9 and 35? There are no corresponding entries in the second matrix that can be added to these entries in the first matrix. So here’s the answer:

We can’t add these matrices A and B, because these matrices are not of the same size.

Question 3. Suppose X, Y, Z, W, and P are matrices of the given order 2 × n, 3 × k, 2 × p, n × 3, and p × k, respectively. The restriction on n, k, and p so that PY + WY can be defined as-

k is arbitrary, p = 2
p is arbitrary, k = 3
k = 2, p = 3
k = 3, p = n

Solution: In this, the order of matrix P = p × k, order of W = n × 3, order of matrix Y = 3 × k. Thus, the order of PY = p×k, when k is equal to 3. And the order of WY = p × k, where p = n. Thus, option (D).

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