To prove that a matrix B is the inverse of a matrix A, you need only use the definition of matrix inverse.

Recall, a matrix B is the inverse of a matrix A if we have

AB=BA=I,

where I is the identity matrix.

So to show that

B=1/(ad−bc)[d, -b; −c, a] is the inverse of

A=[a, b; c, d]

you must perform the matrix multiplication AB and BA and show that both expressions simplify to the 2×2 identity matrix:

I=[1, 0; 0, 1].

In fact, it turns out that if a square matrix is a left (resp. right) inverse, it will also be a right (resp. left) inverse.

If you are trying to determine what B is given A, without any type of formula, I think the easiest way to determine this would be to perform row reduction on the system:

[a, b, 1, 0; c, d, 0, 1].

Assuming det A=ad−bc≠0 (remember, a matrix has no inverse of the determinant is zero), after reducing the first two columns to the 2×2 identity matrix, the last two columns in the system will be your inverse matrix B.

Edit: To better explain why the above method actually works: the augmented system

[a, b, 1, 0; c, d, 0, 1]

represents the matrix equation

AB=I

Here, A is known, and B is unknown.

When we apply an elementary row operation to the system to reduce A to the identity, it is equivalent to left-multiplying the equation AB=I by an elementary matrix. After finitely many row operations, A is reduced to the identity matrix and we have

En⋯E2E1AB=En⋯E2E1I,

where En⋯E2E1 represent the elementary matrices that act as elementary row operations on A, and so En⋯E2E1A=I (because each of these row operations together reduce A to the identity) and the above equation simplifies to

IB=En⋯E2E1

B=En⋯E2E1.

So B is just a product of elementary matrices which act as row operations to reduce A to the identity matrix.

Furthermore, every revertible matrix can be expressed as a product of elementary matrices.

The Matrix inverse you refer to as above, is known as the Moore-Penrose Inverse or Pseudoinverse of the Matrix A, it is unique for every matrix A and exist even if A is strictly rectangular matrix.

For a given matrix A , the psuedoinverse X satisfy the four Moore-Penrose conditions given below :

1. AXA= A 2. XAX= X

3. (AX)^H = AX 4. (XA)^H= XA , where ^H stands for hermitian conjugate.

You can see easily if A is square and invertible, taking X = inv (A) , inv (A) stands for inverse of A , the four Moore-Penrose conditions mentioned above are satisfied.

Considering a square matrix A (n by n), the singular value decomposition of A is given as

A = U(n by n) × S(n by n) × V^H ( n by n),

where V ^H stands for hermitian conjugate of matrix V.

Here U and V are Unitary matrices, S is diagonal matrix whose diagonal elements are singular values of A in non increasing sequence, i.e. S = diag(s1,s2,...., sn)

for invertible A we have s1,s2,...,sn all non zero and positive.

defining X(n by n) = V × inv(S) × U^H , where inv(S) = diag(1/s1, 1/s2,...., 1/sn), you can see that this X satisfy all four Moore-Penrose conditions, hence, it is unique for A and gives AX = XA = I( n by n), where I is the identity matrix .

Therefore this X is inv (A) for the given A. This is how inv(A) is defined for a square invertible matrix A.

The concept of matrix inverse is very intricately related to the concept of Oblique and Orthogonal projection matrices, generalized inverses of a matrix and singular value decomposition (SVD) of a matrix, the following book provides an excellent discussion on the interrelationships of the three concepts mentioned above, understanding this interrelationships is essential for a deep understanding of the matrix inversion problem.

" Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition "

Authors : Haruo Yanai, Kei Takeuchi , Yoshio Takane

Springer Publication

Hope this is helpful.

Dear Araoluwa Filani

Kindly, check the following links of two videos maybe help you for additional understanding.

https://www.youtube.com/watch?v=_mvsgdfz-Ik

https://www.youtube.com/watch?v=iUQR0enP7RQ

Your question is based on the premises of group theory. An element B is said to be the inverse of an element A if AB = BA = e. Where e is the identity element (in this case an identity matrix). If you want to understand the concept of matrix properly, I will implore you to read more on group structure.