Why on finding inverse of a matrix by elementary transformation we always either use row or coloum transformation not both together?
pls give right answer if you have knowledge of matrix and determinants!
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Answers
Step-by-step explanation:
Can we use both row and column transformation in same question of matrices?
*A2A*
I am not sure that I understand your question. But I am guessing that you are attempting to solve the problem of finding inverse of a given matrix or something similar and wondering whether you can apply both row and column transformation to the matrix in question.
If that is your question, then the answer is NO.
Let us take the case of finding an inverse. Given a matrix A you want to find an X such that,
AX=I
or equivalently,
XA=I
Now, the important thing to note that is that position of X above matters. Why? That's because every row transformation (or column transformation) can be written as a product of the given matrix with another matrix from the left (or right).
That is, if you do a row transformation of the form ( Ri=aRj+bRi ), there exists a matrix E such that the result of the row transformation can be written as EA . The matrix E is known as the elementary row transformation matrix (or elementary column transformation matrix, but the elementary column transformation matrix will be multiplied from the right).
So, if you start with the equation,
AX=I
The problem of finding inverse is finding a whole bunch of row transformation whose corresponding elementary row transformation matrices are E1,E2,…,En such that,
En⋯E2E1A=I
This would imply that,
X=En⋯E2E1
So, you can see that if you start with AX = I, you can only do elementary row transformations because you can multiply matrices with A only from the left and similarly, if you start with XA = I, then you can only do elementary column transformations, because you can multiply matrices with A only from the right.
Therefore, you cannot mix up row and column transformation while solving these problems