In one line
Write \(A=I\,A\,I\); every row operation on \(A\) is also done on the left identity (building \(P\)) and every column operation on the right identity (building \(Q\)), until \(A\) becomes normal form.
Explained simply
Each elementary operation is the same as multiplying by a simple "elementary matrix". Row operations multiply from the left, column operations from the right. If you record all row operations in one matrix \(P\) and all column operations in one matrix \(Q\), then \(PAQ\) is the normal form. Think of \(P\) and \(Q\) as the receipts of all the operations you performed.
Technically, for an \(m\times n\) matrix \(A\) of rank \(r\), there exist non-singular \(P\) (\(m\times m\)) and \(Q\) (\(n\times n\)) with \(PAQ=\begin{bmatrix}I_r&0\\0&0\end{bmatrix}\). \(P\) and \(Q\) are not unique; different sequences of operations give different (equally correct) answers.
How it works
- Write \(A_{m\times n}=I_m\,A\,I_n\).
- Row operation: apply it to \(A\) on the left side and to the left \(I_m\) (the pre-factor) on the right side.
- Column operation: apply it to \(A\) and to the right \(I_n\) (the post-factor).
- Continue until the left side is in normal form. The pre-factor is \(P\), the post-factor is \(Q\).
- Verify by multiplying \(PAQ\).
Key formula
\(PAQ=\begin{bmatrix}I_r&0\\0&0\end{bmatrix}\), with \(|P|\ne0\), \(|Q|\ne0\), \(r=\rho(A)\).
Quick example
\(A=\begin{bmatrix}1&2&3\\2&4&6\\1&1&1\end{bmatrix}\). Row operations \(R_2-2R_1\), \(R_3-R_1\), \(R_2\leftrightarrow R_3\), \(R_2\to-R_2\) applied to \(I_3\) give \(P=\begin{bmatrix}1&0&0\\1&0&-1\\-2&1&0\end{bmatrix}\).
Column operations \(C_2-2C_1\), \(C_3-3C_1\), \(C_3-2C_2\) applied to \(I_3\) give \(Q=\begin{bmatrix}1&-2&1\\0&1&-2\\0&0&1\end{bmatrix}\).
Multiplying, \(PAQ=\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}\), so \(\rho(A)=2\).
Common mistakes
- Applying a row operation to \(Q\) or a column operation to \(P\).
- Applying operations to the identity in a different order from \(A\).
- Not verifying \(PAQ\); a quick multiplication catches most errors.
How MU asks it
A standard 6-mark question: "Find non-singular matrices \(P\) and \(Q\) such that \(PAQ\) is in normal form; hence find the rank of \(A\)" (\(3\times3\) only in the syllabus). Write the three-part layout \(A=IAI\) at every step, label each operation, and verify at the end if time allows.