Find a row-reduced echelon matrix which is row-equivalent to
\[ A = \left[ \begin{array}{c c} 1 & - i \\ 2 & 2 \\ i & 1 + i \end{array} \right]. \]
What are the solutions of \(AX = 0\) ?
Solution
Row reduction gives \[ \begin{bmatrix} 1&-i\\ 2&2\\ i&1+i \end{bmatrix} \sim \begin{bmatrix} 1&0\\ 0&1\\ 0&0 \end{bmatrix}. \]
1. The displayed matrix is a row-reduced echelon matrix row-equivalent to\(A\).
Subtract \(2R_1\) from \(R_2\) and \(iR_1\) from \(R_3\). This gives \[ \begin{bmatrix} 1&-i\\ 0&2+2i\\ 0&i \end{bmatrix}. \] The second column contains a nonzero pivot below the first. Scaling that row to make the pivot \(1\), clearing the second column above and below it, yields the displayed RREF.
2. The only solution of \(AX=0\) is \(X=0\).
Row operations preserve the homogeneous solution set, and the reduced system is \(x_1=0\), \(x_2=0\).