Matrix Addition

What Is A Matrix?

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

  • A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.

  • 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.

Image will be uploaded soon

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 = \[\begin{bmatrix} 1  & 2 & 3\\ 7 & 8 & 9\end {bmatrix}\]  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 = \[\begin{bmatrix} 6  & 8 & 10\\ 10 & 12 & 14\end {bmatrix}\] 

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], 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,C ,A = [aij], B = [bij], 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.



 Questions to be solved-

Question 1) 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 add the 3 and the 6, the 12 and 5, the 4 and the 6, and the11 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 the same size

Question 2) 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-

  1. k is arbitrary, p = 2

  2. p is arbitrary, k = 3

  3. k = 2, p = 3

  4. 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).

FAQ (Frequently Asked Questions)

1. How to add matrices and how do you add numbers to a matrix?

Two matrices can be added or subtracted only if the two matrices have the same dimension; in simpler words, we can say that they must have the same number of rows and columns. The operations like addition or subtraction are accomplished by adding or subtracting corresponding elements of any two given matrices. 

The Addition of a scalar to a matrix can be defined as A+b=A+bJd, with d equal to the dimensions of A. This is commutative and associative, just like any regular matrix addition. Then A+b would be the addition of A and bId and A+B the matrix addition as we know it, only valid for the matrices that have the same dimensions.


2. Can you add a 2x3 and a 3x2 matrix and can you add matrices of different sizes?

The important rule to know is that when we need to add and subtract two or more matrices, we need to first make sure the matrices have the same dimensions. In order words, we can say that we can add or subtract a 2x3 matrix with a 2x3 matrix or a 3x3 matrix with a 3x3 matrix. However, you cannot add a 3x2 matrix with a 2x3 matrix or a 2x2 matrix with a 3x3 matrix.

 

In order to add the two matrices, the matrices must have the same dimensions, else you cannot add any two given matrices. In order to multiply two matrices M and N, the number of columns of M must be equal to the number of rows in matrix N.