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Krystian Karczyński- 21 June 2024
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Definition of Matrix Rank and Its Implications
Let’s assume we have defined the rank of a matrix as: “the number of linearly independent rows and columns in a matrix”. What properties of ranks follow from this definition right from the start?
First, it’s obvious that the rank of a matrix can be: 1, or 4, or sometimes 0. But it will definitely not be: -4, or 
.
Okay, is that all?
But is that all we can derive? Let’s take, for example, the matrix:
This matrix has 3 rows and 6 columns.
Let’s ask ourselves, what can the rank of this matrix be? Can it be 7? It’s clear that it can’t, because if the rank of a matrix is “the number of linearly independent rows and columns”, it can’t be 7 in this case because this matrix doesn’t even have that many rows or columns!
Now a harder question… Can the rank be 6? The matrix does have 6 columns…
The answer is: no. 6 would have to be “the number of linearly independent rows and columns”. 6 can be the number of linearly independent columns (because there are 6), but it can’t be the number of linearly independent rows (because there are only 3). And it should be the number of linearly independent “rows and columns”.
So, it’s obvious that the rank of this matrix can be at most 3.
We arrive at a useful property:
rank(A) <= min(number of rows in the matrix, number of columns in the matrix)
Therefore, by looking at a matrix, you can immediately tell its maximum rank – which can sometimes be very useful.
To calculate it more precisely, you need to use appropriate methods – I show them in my Matrix Course in Lesson 5, welcome!
