Matrix Addition

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

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 add 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-

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).