91Ó°ÊÓ

Write down the 3 by 3 matrices that produce these elimination steps: (a) \(-E_{2 t}\) subtracts 5 times row 1 from row 2 . (b) \(E_{32}\) subtracts \(-7\) times row 2 from row 3 . (c) \(P\) exchanges rows 1 and 2 , then rows 2 and 3 .

When zero appears in a pivot position, \(A=L U\) is not possible! (We need nonzero pivots \(d, f, i\) in \(U\).) Show directly why these are both impossible: $$ \left[\begin{array}{ll} 0 & 1 \\ 2 & 3 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ \ell & 1 \end{array}\right]\left[\begin{array}{ll} d & e \\ 0 & f \end{array}\right] \quad\left[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 1 & 2 \\ 1 & 2 & 1 \end{array}\right]=\left[\begin{array}{lll} 1 & & \\ \ell & 1 & \\ m & n & 1 \end{array}\right]\left[\begin{array}{lll} d & e & g \\ & f & h \\ & & i \end{array}\right] $$

Prove that a matrix with a column of zeros cannot have an inverse.

If \(E\) adds row 1 to row 2 and \(F\) adds row 2 to row 1 , does \(E F\) equal \(F E\) ?