A matrix is an array of numbers where it has rows and columns which shows the size or dimensions of the matrices.
For multiplication of the matric by just a single number is very easy –
The calculations are done with the below formula –
We all know that the number 2 in this condition a scalar, and so it is known as the scalar multiplication.
Multiplying the Matrix by another Matrix
But for multiplying the matrix by another matrix we need to solve the dot product of rows and columns and what does it mean? Now let us see an example of it for working out the answer or solution for the 1st row and 1st column.
The dot product is the multiplication of matching members and then the summing up –
(1, 2, 3). (7, 9, 10) = 1 * 7 + 2 * 9 + 3 * 10 = 7 + 11 + 30 = 48
We must match out the 1st members that are 1 and 7, then multiply them for the 2nd members, like 2 and 9 and the 3rd members which are 3 and 10, and then finally sum it all up.
Wish to solve another example? Then here it is for the 1st row and 2nd column –
(2, 3, 4). (7, 8 ,9) = 2 * 7 + 3 * 8 + 4 * 9 = 14 + 24 + 36 = 74
Now we can do same thing for the 2nd row and 1st column –
(1, 2, 1). (2, 3, 4) = 1 * 2 + 2 * 3 + 1 * 4 = 2 + 6 + 4 = 12
And then for the 2nd row and 2nd column –
(3, 1, 2). (1, 2, 3) = 3 * 1 +1 * 2 + 2 * 3 = 3 + 2 + 6 = 11
Then we finally get the result.
Hope you are now clear with the method and solutions?
And what is all about division? We do not actually divide matrices, as we do it in this way.
X / Y = X * (1/Y) = X * B-1
Where, B inverse means the inverse of matrix B. so here we do not need to divide it, but we multiply it by the inverse of a matrix. There are some special ways of finding out the inverse of the matrix.
For the matrix inverse, you need to write out the matrix and identity the matrix side by side and do some of the row operations for making the matrix B identity the matrix. By doing row operations, B will then become the identity matrix and the actual identity matrix will become the inverse of matrix B, as all the operations will be done in the identity matrix.
After getting the matrix inverse you need to multiply the inverse of B with A which will be a division of matrices.
Addition and Subtraction
First of all, let us find out what is the matrix. The matrix can only be added or subtracted to another matrix if both the matrices have equal dimensions. For adding two matrices, just add the entries and have the sum in the place of the resulted matrix.
Let us solve some of the examples and understand it in depth –
Example – 1
[1 2] + [2 -3]
First, notice that both the matrices are 1 * 1 matrices, then we can add it.
[1 2] + [2 -3]
= [1+2 2+(-3)]
= [3 -1]
Subtraction is also a very straightforward procedure with the matrices. Let us look at some examples so that we can have a clear idea about them.
Example – 2
[4 5] – [2 1]
First, clearly see that both the matrices are of the same dimension and then start subtracting –
[4 5] – [2 1]
= [4-2 5-1]
= [2 4]
Addition of matrix is very simple, and it is done with every entry.
Let us solve some critical examples which will give a better understanding of the matrices:
Add the below matrices
Now, here we just need to add this pair of entries and then simplify the final solution.
So, the final answer is –
Until now, you have learned how to add two things in matrices such as variables, numbers, equations amongst others. But addition does not always work with matrices.
Evaluate the below task, or if it is not possible, then give the reason for it.
Although matrices are added with every entry, we need to add two numbers like 2 and 2, 1 and 8, then 3 and 4, 4 and 6. But what else can we do when adding the numbers 6 and 7 and which have no straight numbers in the other matrix? So, the answer is –
These matrices cannot be added, as they do not have the same size and dimensions.
This is always the case when adding the matrices, you need both the matrices of the same dimensions. If they are not of equal sizes, then the addition is not applicable. It does not make any mathematical logic for adding the nonequal matrices.
Subtraction also works with every entry and with same conditions applied. Matric subtraction, as well as addition, can’t be done if the matrices are not of the same dimension or sizes. This is the case for both addition and subtraction of matrices.
Find out the values of x and y for the following equations –
Firstly, you need to evaluate the left-hand side easily with side and entry wise –
Thus, with the equality of matrix works with entry wise, we compare these entries for creating the simple equations that we can solve. In such cases,
X + 6 = 7 and 2y -3 = -5
X = 7 – 6
X = 1
2y – 3 = -5
Y = -5 + 3 / 2
Y = -2 / 2
Y = -1