12:59 PM
3
What does it mean if det(A) equals 1? Does it mean that the identity matrix can be obtained from A by only adding multiples of rows onto others?
The OP asks whether every matrix such that $\det(A)=1$ can be transformed to identity matrix using only adding multiple of some row to another row.
For example the linear combination with coefficients $b_{11},b_{12},...,b_{1n}$ yields the row $(1,0,....,0)$.
However, we are only allowed to replace the first row by a linear combination with coefficients $1,a_2,a_3,\dots,a_n$. Similarly for the second rows we are only allowed to use $b_1,1,b_3,\dots,b_n$.
This would correspond to multiplying from the left by a matrix which has all diagonal entries equal to $1$.
1:48 PM
Even if we solve somehow the problem with diagonal elements, the solution I have suggested above still has gaps.
I argued that $BA$ can be obtained from $A$ using linear combinations where coefficients are determined by the rows of the matrix $A$.
2:07 PM
next day → last day (3672 days later) »
Transcript for
Mar '149
Mar14
Linear & Abstract algebra
For any discussion concerning linear, abstract or even element...