Addition Of Matrices - MathsTips.com

In this article, we will do a matrix addition. Addition between two matrixes is feasible if and only if they have the same order or the dimensions that is they must have the same number of rows and columns. Addition if the matrix is feasible by adding the element of another matrix if they have the same order.

Suppose Mohan has two shops at places A and B. Each shop sells shoes for men and women in three different price categories. The quantities sell by each shop are represented as matrices given below:

Shop 1: $Men\begin {bmatrix} 60\\65 \\85 \end {bmatrix}$ $Women\begin{bmatrix} 50\\55 \\75 \end{bmatrix}$

Shop 2: $Men\begin {bmatrix} 80\\75 \\90 \end {bmatrix}$ $Women\begin{bmatrix} 90\\45 \\85 \end{bmatrix}$

Suppose Mohan wants to know the total sales of shoes in each price categories. So, this can be represented in the matrix form as $Men+Women=\begin {bmatrix}60+80 &50+90 \\ 65+75 &55+45 \\ 85+90 &75+85 \end {bmatrix}$

This new matrix is the sum of the above two matrices. So, the sum of two matrices is obtained by adding the corresponding elements of the given matrices.

Thus if $A=\begin {bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\end {bmatrix}$ is a 2*2 matrix and $B=\begin {bmatrix} b_{11} & b_{12}\\ b_{21} & b_{22}\end {bmatrix}$ is another 2*2 matrix. Then, we define $A+B=\begin {bmatrix} a_{11}+b_{11} & a_{12}+b_{12}\\ a_{21}+b_{21} & a_{22}+b_{22}\end {bmatrix}$

In general, if matrix $A=[a_{ij}]$ and matrix $B=[b_{ij}]$ are the two matrices of the same order, say m*n.

Thus, the sum of two matrices of A and B is defined as a matrix $C=[c_{ij}]$ where $C[c_{ij}]=A[a_{ij}]+B[b_{ij}]$ for all the values of i and j.

Properties of addition of matrices

1. Commutative Law

It means that if $A=[a_{ij}]$ , $B=[b_{ij}]$ are matrices of the same order, say m*n, then A+B=B+A
Thus, A+B= $A[a_{ij}]+B[b_{ij}]$

B+A= $B[b_{ij}]+A[a_{ij}]$

2. Associative Law

It means that 3 matrices that is $A=[a_{ij}]$ , $B=[b_{ij}]$ , $C=[c_{ij}]$ are matrices of same order, say m*n then (A+B) +C=A+ (B+C).

Thus, (A+B) +C= $(A[a_{ij}]+B[b_{ij}])$ + $C[c_{ij}]$
= $A[a_{ij}]$ + $(B[b_{ij}]+C[c_{ij}])$ = A+ (B+C)

Conditions for Addition of matrices

Two matrices should be of same order (number of rows=number of columns).
Add the corresponding element of other matrices.

Addition of matrix of order 2*2

Example 1: Given $A=\begin {bmatrix} 8 & 6\\ 5 & 4\end {bmatrix}$ and $B=\begin {bmatrix} 2 & 0\\ 1& 3\end {bmatrix}$ Find A+B.

Solution: $A+B=\begin {bmatrix} 8+2 & 6+0\\ 5+1& 4+3\end {bmatrix}$
$A+B=\begin {bmatrix} 10 & 6\\ 6& 7\end {bmatrix}$

Addition of matrix of order 3*3

Example 2: $A=\begin {bmatrix} \sqrt{5} & 6 &2 \\ 8&4 &7 \\ 3 &12 &-5 \end {bmatrix}$ and $B=\begin {bmatrix} 2 & \sqrt{5}&1 \\ 4&-2 &3 \\ 0&7 &8 \end {bmatrix}$ Find A+B.

Solution: $A+B=\begin {bmatrix} 2+\sqrt{5} & 6+\sqrt{5}&3 \\ 12&2 &10 \\ 3&19 &3 \end {bmatrix}$

Exercise

Let $A=\begin {bmatrix} 1 & 3\\ -2 & 5\end {bmatrix}$ and $B=\begin {bmatrix} 2 & 3\\ 4 & 2\end {bmatrix}$ Find A+b.
Let $A=\begin {bmatrix} \sqrt{3} & 1 & -1 \\ 2 & 3 & 0 \end {bmatrix}$ and $B=\begin {bmatrix} 2 & \sqrt{5} & 1 \\ -2 & 3 & 4 \end {bmatrix}$ Find A+B.
Let $A=\begin {bmatrix} 8 &6 &1 \\ 2 & -5 & 4 \end {bmatrix}$ and $B=\begin {bmatrix} 0 &8 \\ 1&4 \\ 3 &7 \end {bmatrix}$ Find A+B.
Let $A=\begin {bmatrix} 8 & 7\\ -6 & 5\end {bmatrix}$ and $B=\begin {bmatrix} 4 & 1\\ -3 & 13\end {bmatrix}$ and Show that A+B=B+A.
Let $A=\begin {bmatrix} 8 & 12 &5 \\ -3&19 &22 \\ -1 &5 &6 \end {bmatrix}$ , $B=\begin {bmatrix} 1 & 3 &5 \\ -2&6 &7 \\ 11 &13 &-1 \end {bmatrix}$ and $C=\begin {bmatrix} 3 & 6 &8 \\ -8&9 &2 \\ 7 &4 &5 \end {bmatrix}$ Show that (A+B) +C=A+ (B+C).

Properties of addition of matrices

Conditions for Addition of matrices

Addition of matrix of order 2*2

Addition of matrix of order 3*3

Exercise

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