watch: LA - 高山 04 | Rank of matrix

Find the rank

Rank of a matrix 𝐀 equals to the number of the non-zero rows of a matrix in row echelon form (after performing elementary row operations).

determinant

The matrices appearing in the transforming process are able to reach the same row echelon form, so they have the same rank.
In another word, the elementary row operations don’t change the rank of matrices.
The rank of a matrix is no bigger than the number of rows or columns.

Thus, the solution of a linear equation system can be represented as its rank.

Homogeneous linear equation system:

Only zero solution: r(𝐀) = n, the rank of the coefficient matrix equals to the number of unknowns
Exist non-zero solution: r(𝐀) < n,

Non-homogeneous linear equation system:

Single unique solution: r(𝐀|𝐛) = r(𝐀) = n, the rank of the augmented matrix = the rank of the coefficient matrix = the number of the unknowns;
Inifinitely many solutions: r(𝐀|𝐛) = r(𝐀) < n;
No solution: r(𝐀|𝐛) ≠ r(𝐀)

𝐀 = $[^{\_{0\ 0\ 0}} \_{^{0\ 0\ 0} \_{0\ 0\ 0}}]$

All of its elements are 0. There is no non-zero rows. So r(𝐀) = 0.

This is the so-called zero matrix noted as 𝐀 = 𝟎

r(𝐀) = 0 and 𝐀=𝟎 are equivalent, because if any one of elements is not 0, then there is a non-zero row resulting the r(𝐀) ≠ 0.

𝐀 = $[^{\_{1\ 2\ 3}} \_{^{2\ 4\ 6}\_{3\ 6\ 9}}]$ ➔ $[^{\_{1\ 2\ 3}} \_{^{0\ 0\ 0}\_{0\ 0\ 0}}]$

The rows in the matrix of r(𝐀) = 1 are proportional to each other. In fact, the columns are also proportional.
Their ratio factor can be 0, and also there must be at least 1 element which is not zero in the matrix. Otherwise, it’s the zero matrix.

𝐚= [1 2 3] and 𝐛 = $[^{_1} \_{^2\_3}]$ are 1x1 matrix with rank = 1.

Vectors are the matrix with only 1 row or 1 column.
Row vector with at least 1 non-zero value is of rank=1.
Column vector has multiple rows with a single value. If it’s not a zero vector, the later rows can be reduced to 0 by adding the row1 multiplying with different factors. 𝐛 = $[^{_1} \_{^2\_3}]$ ➔ $[^{_1} \_{^1\_1}]$
For all the non-zero vector, no matter row vector or column vector, their rank = 1.
Vector 𝐚 ≠ 0 and r(𝐚) = 1 are equivalent.