Properties of Matrices Inverse
Matrix Inverse Explained
Before heading to the matrix inverse properties, it is crucial to first understand the meaning and mechanism of the matrix as well the inverse of a matrix. That said, Matrices are robust mathematical tools that can be used in making computer games and all the exciting stuff that appears on the computer screen. Besides, Computer gaming and game graphics, it also has a huge array of applications that ranges from:

Huge data visualization,

Structuring buildings resistant to earthquakes or natural calamities

Measuring infant mortality and beyond
Definition of Matrix and the Inverse of a Matrix
Matrices, although are complex to use manually, are very useful when incorporated through computers and. Matrices are capable of solving cumbersome problems very easily and quickly.
There are ways to add, subtract and multiply matrices, but there is no concept of division of a matrix operation. And this is where the inverse of a matrix comes into action. The nearest we can reach to division of a matrix is—multiplying by its inverse.
For all you know from functions with integer numbers that dividing by a number gets you the same answer as multiplying by its reciprocal. 20/10= 10 x (1/10) = 2. The same holds truth for the matrix inverses – until that matrix has an inverse.
Inverse Matrix Properties
There are a couple of inverse properties to take into account when talking about the inverse of a matrix. First, if multiplying a matrix by its inverse, the sequence does not matter. This is largely atypical for matrix functions because XZ barely equals ZX for the majority of matrices. Second, the inverse of a matrix may not even exist. In case the determinant of a matrix is zero, you just can NOT divide by that!
There Are Basically 3 Other Properties Of The Inverse As Below:
1. If X is a square matrix and Z is the inverse of X, then X is the inverse
Of Z, since XZ = I = ZX. Then we acquire the identity inverse:
(X−1)−1 = X
2. Observe that Z−1X−1XZ = B−1
IZ = I = XZZ−1X−1
. Then:
(XZ)−1 = Z−1X−1
Then in a similar manner as transpose, taking the inverse of a product reverse the sequence of the product.
3. Finally, bear in mind that (XZ)
T = ZT XT
. Since I
T = I, then (X−1X)
T =XT (X−1) T = I.
Likewise, (XX−1) T = (X−1) T XT = I.
Then:
(X−1) T = (XT)−1
Therefore, we could even M−1V write X−T for the inverse transpose of X
(Image to be added soon)
Need of an Inverse
Since with matrices there is no such operation of division, thus we don’t divide by a matrix. However, we can multiply by an inverse, which accomplishes the same purpose.
Solved Example
Just Imagine we cannot divide by numbers and someone asks “How do I distribute 20 mangoes amongst 4 people?” But we can still use the reciprocal of 4 (which is 0.25), so we answer:
Likewise the same theory can be applied with matrices:
Say we want to find matrix O, and we know matrix M and N:
OM = BN
It would be good to divide both sides by M (to obtain O=N/M), but recall we can’t divide.
But, what if we multiply both sides by M1 ?
OMM1 = NM1
And we notice that MM1 = I, so:
OI = NM1
Here, we can eliminate I (for the same logic we can eliminate “1” from 1^{o} = mn for numbers):
O = NM1
And we get our answer (supposing we can calculate A1)
MN is almost never equal to NM.
Calculate the Inverse of a 2×2 Matrix Operations
To obtain the inverse of a 2×2 matrix, you will require following a few steps:
Swap the numbers in (row 1 & column 1) and (row 2 & column 2)
Give opposite signs to the numbers in (row 1 & column 2) and (row 2 & column 1)
Now, finally divide by the determinant of the native matrix
Fun Facts

The Inverse May Not Exist

With matrices the sequence of multiplication matters

Firstly, to get an inverse the matrix should be “square” of same number of rows and columns

The determinant cannot be zero (or we finish as dividing by 0)

Only computer can find inverses to most real systems of equations that are very big (up to 100 by 100 is)
1. Why Do We Need Matrices And Determinants?
Matrices are quite crucial for solving huge sets of corresponding equations by using a digital system (computer). Beyond a doubt, we don’t want to consider a different letter for each variable in our mathematical problem (or tons of subscripts,) as it would delay the solution process and would make coding terrible. With matrices, we are not needed to include any variables – only the numbers in front of those variables.
2. How To Solve 4×4 System Of Equations
Therefore for example, if we are attempting to solve this 4×4 system of equations:
2a + 3b + 1c −5z = 4
a – 4b + 6c + 9c = −7
7a + 4b − c + 6w = 7
5a – 3b + 11c + 14w = 3
We are only required to provide the computer the coefficients, like this:
2 3 1 −5  4
1 −4 6 9  −7
7 4 −1 6  7
5−3 11 14  3
The computer only operates on the numbers − it doesn’t require the letters.
3. What Is An Identity Matrix?
Identity matrix is the same number of rows and columns. It has ‘1’ in every mark of the diagonal running down from upper left to lower right, while it has ‘0’ everywhere else.
Identity matrices can be in any size required: 5×5, 15×15, even 500x 500. 3×3 is the standard size of matrix usually used in computer graphics, but apply them to tens of thousands of separate points that reconcile the weapons, devils, landscape, you communicate with on the screen.