In one line
The rank of a matrix is the order of its largest non-zero minor, which equals the number of linearly independent rows; elementary transformations do not change the rank, so we use them to simplify the matrix first.
Explained simply
Suppose a class has three survey questions, but question 3 is just "question 1 + question 2". It adds no new information. The rank counts how many rows carry genuinely new information. A \(3\times3\) matrix whose third row is the sum of the first two has rank 2, not 3.
Technically, a minor of order \(r\) is the determinant of an \(r\times r\) submatrix. The rank \(\rho(A)=r\) if at least one \(r\)th-order minor is non-zero and every minor of order \(r+1\) (if any) is zero. The rank of an \(m\times n\) matrix is at most \(\min(m,n)\); only the zero matrix has rank 0.
How it works
Elementary transformations (row or column):
- Interchange two rows: \(R_i\leftrightarrow R_j\).
- Multiply a row by a non-zero constant: \(R_i\to kR_i\).
- Add a multiple of one row to another: \(R_i\to R_i+kR_j\).
The same three operations on columns are written with \(C\). Matrices connected by such operations are called equivalent (\(A\sim B\)) and have the same rank. We use them to create zeros until the rank is obvious (echelon or normal form).
Key formula
\(\rho(A)=r\) if some \(r\times r\) minor \(\ne0\) and all \((r+1)\times(r+1)\) minors \(=0\). \(\rho(A)\le\min(m,n)\) for an \(m\times n\) matrix; \(\rho(A)=n\) for an \(n\times n\) matrix iff \(|A|\ne0\).
Quick example
\(A=\begin{bmatrix}1&2&3\\2&4&6\\1&1&1\end{bmatrix}\).
\(|A|=0\) because row 2 is twice row 1. So the rank is less than 3. The \(2\times2\) minor from rows 1, 3 and columns 1, 2 is \(\begin{vmatrix}1&2\\1&1\end{vmatrix}=1-2=-1\ne0\). Hence \(\rho(A)=2\).
Finding rank by minors becomes slow for bigger matrices (a \(4\times4\) matrix has 16 third-order minors), which is why the echelon method is preferred.
Common mistakes
- Concluding the rank is 1 just because \(|A|=0\); check the \(2\times2\) minors.
- Using a non-elementary operation like \(R_1\to R_1\times R_2\) or multiplying a row by 0.
- Thinking the rank can exceed the number of rows or columns.
How MU asks it
The definition can be a 3-mark part ("Define rank of a matrix"), but usually it leads into "Find the rank by reducing to echelon/normal form" (5/6 marks). State the definition with minors, write each elementary operation used beside the arrow, and give the final rank with a one-line reason.