In Gaussian elimination, what is a \(pivot\)?

What does an inconsistent row such as \((0,0\mid5)\) mean?

Why might Gaussian elimination require swapping rows before choosing a pivot?

In triangular form, do elimination steps create zeros above or below \(pivots\)?

Use back substitution to solve \(z=2\), \(y+3z=10\), \(x-y+z=1\).

For \(\left(\begin{array}{cc|c}1&2&5\\3&7&16\end{array}\right)\), perform one elimination step to create a zero below the first pivot.

Solve the triangular system \(2x-y+z=5\), \(3y+2z=13\), \(z=2\).

The first column of an augmented matrix begins \((0,2,-1)^T\). What should you do before using the first pivot?

Use Gaussian elimination to solve \(x+y=6\), \(2x+3y=15\).

Use Gaussian elimination to solve \(x+y+z=6\), \(2x+y-z=3\), \(x-y+2z=7\).

During elimination, a row becomes \((0,0,0\mid0)\). Explain how this affects the solution process.

Use elimination to show that \(x+y=2\), \(2x+2y=5\) is inconsistent.

Solve \(2x+y-z=1\), \(4x+3y+z=7\), \(-2x+y+2z=4\) by Gaussian elimination.

A system has triangular augmented matrix \(\left(\begin{array}{ccc|c}1&2&-1&0\\0&0&3&6\\0&0&0&0\end{array}\right)\). Describe the solution set.

For what values of \(k\) does elimination on \(x+y=1\), \(2x+2y=k\) produce an inconsistent row?

For which values of \(a\) does the first elimination step for \(ax+y=1\), \(2x+3y=4\) need a row swap if row \(1\) is chosen first?

Classify the solutions of \(x+y+z=1\), \(2x+2y+2z=2\), \(x-y=0\) using Gaussian elimination reasoning.

Diagnose the error: after reaching \(\left(\begin{array}{ccc|c}1&2&1&5\\0&1&-1&0\\0&0&2&6\end{array}\right)\), a learner solves from the top row first.

Explain why a zero row \((0,0,0\mid0)\) and an inconsistent row \((0,0,0\mid1)\) must not be treated the same way.

Give a strategy argument for why Gaussian elimination preserves the solution set when it uses only \(elementary\) row operations.