Let us see about inverse of a square law maths. Let A is the four-sided figure(square) matrix of order n. If there subsist the matrix B of order n so that AB = BA = I, Here I is the identity matrix of order n, after that the matrix A is thought to be invertible and B is known as the inverse or reciprocal of A.
Note
Note 1:
Only a square matrix has its inverse.
Note 2:
If B is the inverse of A, then A is the inverse of B.
Note 3:
Inverse of A is represented by A-1, thus B = A-1 and AA-1 = A-1A=I.
Theorems
The following is the theorems of inverse square law maths.
Theorem 1
In square law maths, inverse of a square matrix if it subsists is unique
Proof:
Let A is a square(four-sided figure) matrix. If feasible, let B and C is a two inverse of A.
So,
AB = BA = I.
AC = CA = I
Now,
B = BI = B(AC)
= (BA)C [ Matrix multiplication is associative]
= IC = C
i.e., B = C
Therefore the inverse of A is unique.
I am planning to write more post on Converting Percents to Fractions with example, online math tutor free. Keep checking my blog.
Theorem 2
Prove in square law maths, If A and B are the 2 invertible matrices of equal order, in that case (AB)-1 = B-1A-1.
Proof:
From the inverse of a matrix, we have
(AB)(AB)-1 = I
or A-1 (AB)(AB)-1 = A-1 I (Pre multiplying equally sides by A-1)
or (A-1A) B (AB)-1 = A-1 (Since A-1 I = A-1)
or I B (AB)-1 = A-1
or B (AB)-1 = A-1
or (B-1B)(AB)-1 =B-1A-1
or I(AB)-1= B-1A-1
or (AB)-1 = B-1A-1
Theorem 3
Properties of Inverse of Matrix
The following is the properties of inverse square law maths
In other way, a square matrix A is invertible if and only if A is a not a singular matrix.
(c) If A and B are invertible square matrices, then
(AB)-1 = B-1 A-1
(d) If A and B are two plural square matrices of the similar order, then AB and BA are also not a singular matrices of the similar order.
Get Math Help 