1.
Let \(A\in M_{n\times n}(F)\text{.}\) Show that
\begin{equation*}
e(I_n)\cdot A=e(A).
\end{equation*}
Exercises 1.2 Exercises
Let \(e\colon M_{n\times n}(F)\to M_{n\times n}(F)\) be an elementary row operation (of any type).
2.
Keep the notations of the above exercise. If we multiply \(A\) by \(e(I_n)\) on the right, i.e., if we consider the matrix \(A\cdot e(I_n)\) then, what matrix one gets? Relate this with "elementary column operations".3.
Keep notations of the above exercise. Compute determinant of \(e(I_n)\) for each type of elementary row operations. Is \(e(I_n)\) invertible matrix? Justify your answer.
Hint.
Let us denote by
- \(T_{pq}(\alpha)\) the matrix obtained by multiplying \(q\)-th row of \(I_n\) by \(\alpha\in F\) and adding this row to \(p\)-th row of \(I_n\text{;}\)
- \(D_p(\beta)\) the matrix obtained by multiplying \(p\)-th row of \(I_n\) by \(\beta\in F\setminus\{0\}\text{;}\)
- \(P_{rs}\) be the matrix obtained by interchanging \(r\)-th row with \(s\)-th row of \(I_n\text{.}\)
Check that
- \(T_{pq}(\alpha)^{-1}=T_{pq}(-\alpha)\text{;}\)
- \(D_p(\beta)^{-1}=D_p(\beta^{-1})\text{;}\)
- \(P_{pq}^{-1}=P_{pq}\text{.}\)
4.
Is "row-equivalence" an equivalence relation?5.
Let \(A,B\in M_{m\times n}(F)\) be matrices over a field \(F\text{.}\) Assume that homogeneous systems of linear equations \(AX=0\) and \(BX=0\) have same solutions. Is it true that \(A\) and \(B\) are row-equivalent? [Note that this was a question asked by one of the student.]
Hint.
See Section 1.4
