Test: Elementary Operations

An n×n matrix A is said to be invertible if there exists an n×n matrix B such that AB=I, and BA=I, where I is the n×n identity matrix.
If such a matrix B exists, then it is known to be unique and called the inverse matrix of A, denoted by A⁻¹.

The inverse exists only for square matrix. However this is not true. A nonsingular matrix must have their inverse whether it is square or nonsquare matrix.

A = IA
AA^-1 = I
A = {(1,0),(-1,1)}
A^-1{(1,0),(-1,1)} =  {(1,0),(0,1)}
R2 ----> R2 + R1
A^-1 {(1,0),(0,1)} = {(1,0),(1,1)}
I A^-1 = {(1,0),(1,1)}
Therefore, A^-1 = {(1,0),(1,1)}

A = {(0,1) (1,0)}
Adj(A) = {(0,(-1)) ((-1),0)}
|A| = (0 - 1) = -1
A^-1 = adj(A)/|A|
= -1{(0,(-1)) ((-1),0)}
= {(0,1) (1,0)}

ANSWER :- a

Solution :- Matrix equivalence is an equivalence relation on the space of rectangular matrices.

For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions

• The matrices can be transformed into one another by a combination of elementary row and column operations.

If a matrix B is obtained from matrix A by an elementary row or column transformation then B is said to be equivalent of A.​