What is the normal form of a matrix and how do I reduce a matrix to it to find rank?

In one line

The normal form of a matrix is \(\begin{bmatrix}I_r&0\\0&0\end{bmatrix}\), obtained with both row and column operations; the size \(r\) of the identity block is the rank.

Explained simply

Echelon form is a staircase. Normal form goes further: you clean up everything except a neat block of 1s on the diagonal, like tidying a cupboard until only the essentials remain in a straight line. Counting those 1s gives the rank.

Technically, every non-zero \(m\times n\) matrix of rank \(r\) can be reduced by elementary row and column operations to one of the forms \(I_r\), \([I_r\ \ 0]\), \(\begin{bmatrix}I_r\\0\end{bmatrix}\) or \(\begin{bmatrix}I_r&0\\0&0\end{bmatrix}\). This is the normal (canonical) form. Unlike echelon form, column operations are allowed here.

How it works

1. Use row operations to create zeros below the leading entries (as in echelon form).
2. Use column operations \(C_j\to C_j-kC_1\) to clear the rest of the first row.
3. Repeat for the second row/column, and so on.
4. Scale rows or columns to make leading entries 1.
5. Read \(r\) from the identity block.

Key formula

\(A\sim\begin{bmatrix}I_r&0\\0&0\end{bmatrix}\Rightarrow\rho(A)=r\), where \(I_r\) is the \(r\times r\) identity matrix.

Quick example

\(A=\begin{bmatrix}1&2&3\\2&4&6\\1&1&1\end{bmatrix}\).

Rows: \(R_2-2R_1\), \(R_3-R_1\), \(R_2\leftrightarrow R_3\), \(R_2\to-R_2\): \(\begin{bmatrix}1&2&3\\0&1&2\\0&0&0\end{bmatrix}\).

Columns: \(C_2-2C_1\), \(C_3-3C_1\): \(\begin{bmatrix}1&0&0\\0&1&2\\0&0&0\end{bmatrix}\); then \(C_3-2C_2\): \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}=\begin{bmatrix}I_2&0\\0&0\end{bmatrix}\).

So \(\rho(A)=2\).

Common mistakes

• Stopping at echelon form when normal form is asked.
• Writing a column operation as \(R\) or vice versa, which confuses the examiner and matters in PAQ problems.
• Leaving a diagonal entry as \(-1\) or 2; normal form needs 1s.

How MU asks it

Usually a 5/6-mark part: "Reduce \(A\) to normal form and find its rank" for a \(3\times3\) matrix (the \(m\times n\) case is self-learning). Show each operation clearly labelled R or C, and display the final block form with the rank stated.