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Rank of a Matrix

Reducing a Matrix (Gauss-Jordan Elimination)Previous Section

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Rank of a Matrix
The rank of a matrix AAA , denoted r=rank(A)r = {\rm rank}(A)r=rank(A) , is the number of leading 1s in the RREF of the matrix.
Example
Given [A ∣ b⃗ ]=[101001200001]\left[A \ |\ \vec{b}\  \right]
=
\left[\begin{array}{ccc|c}
1&0&1&0\\
0&1&2&0\\
0&0&0&1
\end{array}\right][A ∣ b ]=​100​010​120​001​​,
r(A)=r(A)=r(A)= 2
r([A ∣ b⃗ ])=r([A \ |\ \vec{b} \ ])=r([A ∣ b ])= 3
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Full rank
A matrix has full rank if every column or every row of the RREF of the matrix contains a leading 1.

Note: rank(Am×n)≤min⁡(m,n){\rm rank}(A_{m \times n}) \le \min(m,n)rank(Am×n​)≤min(m,n)
Example
Find the maximum rank of the matrix A=[a11a12a13a21a22a23]A =
\begin{bmatrix}
a_{11} & a_{12} & a_{13}\\ 
a_{21} & a_{22} & a_{23}\\ 
\end{bmatrix}
A=[a11​a21​​a12​a22​​a13​a23​​] .

AAA has size  2×32 \times 32×3  so rank(A)≤2{\rm rank}(A) \le 2rank(A)≤2.
In this case, if AAA has full rank, each row must contain a leading 1 (not every column, since the rank can be at most 2). 

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Types of Solutions
A system of linear equations may be:
Inconsistent  (no solution)  ⟺  rank(A)<rank([ A ∣ b⃗ ])\iff {\rm rank}(A) < {\rm rank}([\ A\ |\ \vec b\ ])⟺rank(A)<rank([ A ∣ b ])
Consistent (one solution, or infinitely many solutions)   ⟺  rank(A)=rank([ A ∣ b⃗ ])\iff {\rm rank}(A) \bm= {\rm rank}([\ A\ |\ \vec b\ ])⟺rank(A)=rank([ A ∣ b ])
No Solutions
In the RREF of [A ∣ b⃗ ][A \ |\ \vec b\ ][A ∣ b ], there is a leading entry in the augmented column: [00⋯01]\left[\begin{array}{cccc|c}
0&0&\cdots&0&1
\end{array}\right][0​0​⋯​0​1​]
Exactly One Solution
There is a unique solution   ⟺  rank(A)=rank([ A ∣ b⃗ ])=n\iff {\rm rank}(A) \bm= {\rm rank}([\ A\ |\ \vec b\ ]) \colorTwo{\bm = n}⟺rank(A)=rank([ A ∣ b ])=n.
In the RREF of [A ∣ b⃗ ][A \ |\ \vec b\ ][A ∣ b ]:
There is no leading entry in the augmented column
every column of the coefficient matrix (left) has a leading 1
Infinitely Many Solutions
There are infinitely many solutions   ⟺  rank(A)=rank([ A ∣ b⃗ ])<n\iff {\rm rank}(A) \bm= {\rm rank}([\ A\ |\ \vec b\ ]) \colorTwo{\bm < n}⟺rank(A)=rank([ A ∣ b ])<n.
In the RREF of [A ∣ b⃗ ][A \ |\ \vec b\ ][A ∣ b ]:
There is no leading entry in the augmented column
There is at least one column without a leading 1
This results in an k\colorFour{\bm k}k-parameter family of solutions where k=n−rank(A)k=n-{\rm rank}(A)k=n−rank(A).

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Example: Rank of a Matrix
Find the ranks of the following matrices and state the number of solutions.
[102100∣120]\left[\begin{array}{l}
 1& 0  \\
 2& 1 \\
 0 & 0  \\
 \end{array}\right.\left|\begin{array}{l} 1\\2\\0\end{array}\right]​120​010​​120​​

[101212000]→R2  →  R2−2R1[101010000]\left[\begin{array}{cc|c}
1&0&1\\
2&1&2\\
0&0&0
\end{array}\right]

\xrightarrow{R_2 \ \ \rightarrow \ \  R_2-2R_1}

\left[\begin{array}{cc|c}
1&0&1\\
0&1&0\\
0&0&0
\end{array}\right]​120​010​120​​R2​  →  R2​−2R1​​​100​010​100​​
rank(A)=2{\rm rank}(A)=2rank(A)=2 which is full rank because AAA has only 2 columns.
rank([A ∣ b⃗ ])=2{\rm rank}([A \ |\ \vec{b}\ ])=2rank([A ∣ b ])=2 (there is no leading 1 in the augmented column)
n=2n=2n=2 because this represents a system of equations with 2 unknowns (2 columns)

Since  rank(A)=rank([A ∣ b⃗ ])=n{\rm rank}(A)={\rm rank}([A \ |\ \vec{b}\ ])=nrank(A)=rank([A ∣ b ])=n, this system has a unique solution.

 
[10201−1000∣310]\left[\begin{array}{l}
1 & 0 & 2  \\
0 &1 & -1 \\
0 & 0 & 0  \\
 \end{array}\right.\left|\begin{array}{l} 3\\1\\0\end{array}\right]​100​010​2−10​​310​​

This matrix is already in RREF.
rank(A)=2{\rm rank}(A)=2rank(A)=2 which is not full rank.
rank([A ∣ b⃗ ])=2{\rm rank}([A\ |\ \vec{b}\ ])=2rank([A ∣ b ])=2
n=3n=3n=3 because this represents a system of equations with 3 unknowns (3 columns)
Since rank(A)=rank([A ∣ b⃗ ])<n{\rm rank}(A)={\rm rank}([A\ |\ \vec{b}\ ])<nrank(A)=rank([A ∣ b ])<n, this system has infinitely many solutions.
In particular, since  n−rank(A) = 3−2 = 1n-{\rm rank}(A) \ =\ 3-2 \ =\ 1n−rank(A) = 3−2 = 1, there is a 1-parameter family of solutions.

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Suppose AAA is a 3×53\times 53×5 matrix with rank(A)=2{\rm rank}(A)=2rank(A)=2 and rank([A ∣ b⃗ ])=3{\rm rank}([A\ |\ \vec{b}\ ])=3rank([A ∣ b ])=3.
How many solutions are there for the linear system with coefficient matrix AAA and constant vector b⃗\vec bb?

rank(A)=2{\rm rank}(A)=2rank(A)=2
rank([A ∣ b⃗ ])=3{\rm rank}([A\ |\ \vec{b}\ ])=3rank([A ∣ b ])=3
n=5n=5n=5 (read from the size of Am×n=A3×5A_{m \times \colorTwo{n}} = A_{3 \times \colorTwo{5}}Am×n​=A3×5​)
Since rank(A)<rank([A ∣ b⃗ ]){\rm rank}(A)<{\rm rank}([A\ |\ \vec{b}\ ])rank(A)<rank([A ∣ b ]), this system has no solutions.
That's because, in RREF, the augmented column must contain a leading 1: [00…∣  1]\begin{bmatrix}
  0& 0& \dots &|\ \  1\\
\end{bmatrix}
[0​0​…​∣  1​].
This means that the sum of 0 times every variable equals 1, i.e. 0=10=10=1, which is impossible.

In the reduced row echelon form of the coefficient matrix AAA that corresponds to a system of linear equations with 6 equations in 9 unknowns, the maximum number of leading 1s is 
.

I don't know

Determine the number of solutions for the system of linear equations Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b with 5 equations and 5 unknowns, given:

The rank of AAA is 5 and the rank of [A ∣ b⃗ ][A\ |\ \vec{b}\ ][A ∣ b ] is 5.

No solution

Unique solution

Infinitely many solutions

I don't know

Extra Practice

What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 15

The linear system M‾⋅x⃗=0⃗\bcb{\boldsymbol{ \underline{M} \cdot  \vx = \vz}}M​⋅x=0  is known to have only the trivial solution. If M‾\bcb{\boldsymbol{ \underline{M}}}M​ is a matrix with 8 rows and 7 columns, what is its rank? 

The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l}
1&0&1&1&1\\
0&1&2&4&0\\
0&0&0&0&1\\
0&0&0&0&0
\end{array}\right.\left|\begin{array}{l}1
\\2\\5\\8\end{array}\right]_.[A∣b]​1000​0100​1200​1400​1010​​1258​​.​

Example: Rank of a Matrix

Example:
1. Find the ranks of the following matrices and state the number of solutions.
a.) [102100∣120]\left[\begin{array}{l}
 1& 0  \\
 2& 1 \\
 0 & 0  \\
 \end{array}\right.\left|\begin{array}{l} 1\\2\\0\end{array}\right]​120​010​​120​​

The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l}
1&0&1&1&1\\
0&1&2&4&0\\
0&0&0&0&1\\
0&0&0&0&0
\end{array}\right.\left|\begin{array}{l}1
\\2\\5\\8\end{array}\right]_.[A∣b]​1000​0100​1200​1400​1010​​1258​​.​

What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.

What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.

A linear system of three equations in four unknowns has

The reduced row echelon form of the augmented matrix for a linear system is
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l}
1&0&1&1&1\\
0&1&2&4&0\\
0&0&0&0&1\\
0&0&0&0&0
\end{array}\right.\left|\begin{array}{l}1
\\2\\5\\8\end{array}\right]_.[A∣b]​1000​0100​1200​1400​1010​​1258​​.​
a) What is the rank of matrix A

Equivalent Conditions to Full Rank

a) If AAA is a 5×35\times 35×3 matrix of full rank, and BBB is a 3×73\times 73×7 matrix of rank 2, what are the possible values of rank(AB)rank(AB)rank(AB) ?
b) If AAA is a 3×53\times 53×5 matrix of full rank and BBB is a 5×45\times 45×4 matrix of rank 3, what are the possible values of rank(AB)rank(AB)rank(AB) ?

Find the rank of the following matrix:
[525151021200]\begin{bmatrix}5&2&5&1\\5&1&0&2\\1&2&0&0\end{bmatrix}​551​212​500​120​​

Suppose that a system of mmmlinear equations in nnnvariables is represented by the augmented matrix [A∣b]\left[A\left|b\right.\right][A∣b]. If the system is consistent, which of the following statements is always true?

Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AAAis rrr, what is the maximum value rrrcan be?

Given a system of 3 linear equations in 5 variables, if the rank of the coefficient matrix AAAis 3, which of the following statements best describes the system?

Suppose that a system of mmmlinear equations in nnnvariables is represented by the augmented matrix [A∣b][A|b][A∣b]. If the system is consistent, which of the following statements is always true?

Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AAA is rrr, what is the maximum value rrrcan be?

Given a system of 3 linear equations in 5 variables, if the rank of the coefficient matrix AAA is 3, which of the following statements best describes the system?

Linear equations

Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AAAis rrr, what is the maximum value rrrcan be?

Matrix of a Projection

a) Find the matrix AAA so that for x⃗∈R3\vec x \in \mathbb{R}^3x∈R3 the transformation mapping x⃗→proj(−13,1,12)x⃗\vec x\rightarrow proj_{(\frac{-1}{3},1,\frac{1}{2})}\vec xx→proj(3−1​,1,21​)​x is the same as the transformation mapping  x⃗→Ax⃗\vec x\rightarrow A\vec xx→Ax 
b) What is rank(A)rank(A)rank(A) ?

Invertible Matrix Theorem

Example: Invertible Matrix Theorem
Consider the linear system 𝐴𝑥⃗=𝑏⃗𝐴𝑥⃗=\vec{𝑏}Ax⃗=b where A5×5A_{5 \times 5}A5×5​ is an invertible matrix. What is rank([𝐴 ∣ b⃗]){\rm rank}([𝐴\ |\ \vec{b}])rank([A ∣ b])?

Rank of a Matrix

Determine the number of solutions for the system of linear equations Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b with 5 equations and 5 unknowns, given:

Rank of a Matrix

In the reduced row echelon form of the coefficient matrix AAA that corresponds to a system of linear equations with 6 equations in 9 unknowns, the maximum number of leading 1s is 6
.

Determine the rank of [1−10320131110]\begin{bmatrix}
1&-1&0&3\\
2&0&1&3\\
1&1&1&0
\end{bmatrix}​121​−101​011​330​​

Rank of a matrix

Let AAA be a 6 x 4 matrix and let Ax=bA\bm{x}=\bm{b}Ax=b be a system of linear equations. Which of the following statements must always be true? (select all that apply)

Rank of a Matrix

Determine the rank of [1−10320131110]\begin{bmatrix}
1&-1&0&3\\
2&0&1&3\\
1&1&1&0
\end{bmatrix}​121​−101​011​330​​

Rank of a Matrix

How many solutions does the system of linear equations corresponding to the following augmented matrix have?
[1010−1101101000011−1000000]\left[
\begin{array}{rrrrr|r}
1&0&1&0&-1&1\\
0&1&1&0&1&0\\
0&0&0&1&1&-1\\
0&0&0&0&0&0
\end{array}
\right]​1000​0100​1100​0010​−1110​10−10​​

Rank of a Matrix

Which of the following is an augmented matrix corresponding to a system of linear equation that is consistent? (select all that apply)

Wize University Linear Algebra Textbook > Systems of Linear Equations (SLEs) (Linear Systems)

Rank of a Matrix

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Rank of a Matrix

Full rank

Types of Solutions

Example: Rank of a Matrix

Extra Practice

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 15

Example: Rank of a Matrix

Equivalent Conditions to Full Rank

Linear equations

Matrix of a Projection

Invertible Matrix Theorem

Rank of a Matrix

Rank of a Matrix

Rank of a matrix

Rank of a Matrix

Rank of a Matrix

Rank of a Matrix