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

Rank of a Matrix

Rank of a Matrix

The rank of a matrix $A$ , denoted $r = {\rm rank}(A)$ , is the number of leading 1s in the RREF of the matrix.

Example

Given $\left[A \ |\ \vec{b}\ \right] = \left[\begin{array}{ccc|c} 1&0&1&0\\ 0&1&2&0\\ 0&0&0&1 \end{array}\right]$ ,

$r(A)=$ ________
$r([A \ |\ \vec{b} \ ])=$ ________

Full rank

A matrix has full rank if every column or every row of the RREF of the matrix contains a leading 1.

Note: ${\rm rank}(A_{m \times n}) \le \min(m,n)$

Example

Find the maximum rank of the matrix $A = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ \end{bmatrix}$ .

$A$ has size $2 \times 3$ so ${\rm rank}(A) \le 2$ .

In this case, if $A$ has full rank, each row must contain a leading 1 (not every column, since the rank can be at most 2).

Types of Solutions

A system of linear equations may be:

Inconsistent (no solution) $\iff {\rm rank}(A) < {\rm rank}([\ A\ |\ \vec b\ ])$
Consistent (one solution, or infinitely many solutions) $\iff {\rm rank}(A) \bm= {\rm rank}([\ A\ |\ \vec b\ ])$

No Solutions

In the RREF of $[A \ |\ \vec b\ ]$ , there is a leading entry in the augmented column: $\left[\begin{array}{cccc|c} 0&0&\cdots&0&1 \end{array}\right]$

Exactly One Solution

There is a unique solution $\iff {\rm rank}(A) \bm= {\rm rank}([\ A\ |\ \vec b\ ]) \colorTwo{\bm = n}$ .

In the RREF of $[A \ |\ \vec b\ ]$ :

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 $\iff {\rm rank}(A) \bm= {\rm rank}([\ A\ |\ \vec b\ ]) \colorTwo{\bm < n}$ .

In the RREF of $[A \ |\ \vec b\ ]$ :

There is no leading entry in the augmented column
There is at least one column without a leading 1

This results in an $\colorFour{\bm k}$ -parameter family of solutions where $k=n-{\rm rank}(A)$ .

Example: Rank of a Matrix

Find the ranks of the following matrices and state the number of solutions.
$\left[\begin{array}{l} 1& 0 \\ 2& 1 \\ 0 & 0 \\ \end{array}\right.\left|\begin{array}{l} 1\\2\\0\end{array}\right]$

$\left[\begin{array}{cc|c} 1&0&1\\ 2&1&2\\ 0&0&0 \end{array}\right] \xrightarrow{R_2 \ \ \rightarrow \ \ R_2-2R_1} \left[\begin{array}{cc|c} 1&0&1\\ 0&1&0\\ 0&0&0 \end{array}\right]$

${\rm rank}(A)=2$ which is full rank because $A$ has only 2 columns.
${\rm rank}([A \ |\ \vec{b}\ ])=2$ (there is no leading 1 in the augmented column)
$n=2$ because this represents a system of equations with 2 unknowns (2 columns)

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

$\left[\begin{array}{l} 1 & 0 & 2 \\ 0 &1 & -1 \\ 0 & 0 & 0 \\ \end{array}\right.\left|\begin{array}{l} 3\\1\\0\end{array}\right]$

This matrix is already in RREF.

${\rm rank}(A)=2$ which is not full rank.
${\rm rank}([A\ |\ \vec{b}\ ])=2$
$n=3$ because this represents a system of equations with 3 unknowns (3 columns)

Since ${\rm rank}(A)={\rm rank}([A\ |\ \vec{b}\ ])<n$ , this system has infinitely many solutions.

In particular, since $n-{\rm rank}(A) \ =\ 3-2 \ =\ 1$ , there is a 1-parameter family of solutions.

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

${\rm rank}(A)=2$
${\rm rank}([A\ |\ \vec{b}\ ])=3$
$n=5$ (read from the size of $A_{m \times \colorTwo{n}} = A_{3 \times \colorTwo{5}}$ )

Since ${\rm rank}(A)<{\rm rank}([A\ |\ \vec{b}\ ])$ , this system has no solutions.

That's because, in RREF, the augmented column must contain a leading 1: $\begin{bmatrix} 0& 0& \dots &|\ \ 1\\ \end{bmatrix}$ .

This means that the sum of 0 times every variable equals 1, i.e. $0=1$ , which is impossible.

In the reduced row echelon form of the coefficient matrix $A$ 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

Extra Practice

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)

Rank of a Matrix

How many solutions does the system of linear equations corresponding to the following augmented matrix have?

$\left[ \begin{array}{rrrrr|r} 1&0&1&0&-1&1\\ 0&1&1&0&1&0\\ 0&0&0&1&1&-1\\ 0&0&0&0&0&0 \end{array} \right]$

Rank of a Matrix

Determine the rank of $\begin{bmatrix} 1&-1&0&3\\ 2&0&1&3\\ 1&1&1&0 \end{bmatrix}$

Rank of a matrix

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

Determine the rank of $\begin{bmatrix} 1&-1&0&3\\ 2&0&1&3\\ 1&1&1&0 \end{bmatrix}$

Rank of a Matrix

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

Rank of a Matrix

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

Invertible Matrix Theorem

Example: Invertible Matrix Theorem

Consider the linear system $𝐴𝑥⃗=\vec{𝑏}$ where $A_{5 \times 5}$ is an invertible matrix. What is ${\rm rank}([𝐴\ |\ \vec{b}])$ ?

Matrix of a Projection

a) Find the matrix $A$ so that for $\vec x \in \mathbb{R}^3$ the transformation mapping $\vec x\rightarrow proj_{(\frac{-1}{3},1,\frac{1}{2})}\vec x$ is the same as the transformation mapping $\vec x\rightarrow A\vec x$

b) What is $rank(A)$ ?

Linear equations

Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix $A$ is $r$ , what is the maximum value $r$ can be?

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

Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix $A$ is $r$ , what is the maximum value $r$ can be?

Suppose that a system of $m$ linear equations in $n$ variables is represented by the augmented matrix $[A|b]$ . If the system is consistent, which of the following statements is always true?

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

Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix $A$ is $r$ , what is the maximum value $r$ can be?

Suppose that a system of $m$ linear equations in $n$ variables is represented by the augmented matrix $\left[A\left|b\right.\right]$ . If the system is consistent, which of the following statements is always true?

Find the rank of the following matrix:

\begin{bmatrix}5&2&5&1\\5&1&0&2\\1&2&0&0\end{bmatrix}

Equivalent Conditions to Full Rank

a) If $A$ is a $5\times 3$ matrix of full rank, and $B$ is a $3\times 7$ matrix of rank 2, what are the possible values of $rank(AB)$ ?

b) If $A$ is a $3\times 5$ matrix of full rank and $B$ is a $5\times 4$ matrix of rank 3, what are the possible values of $rank(AB)$ ?

The reduced row echelon form of the augmented matrix for a linear system is

$[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.$

a) What is the rank of matrix A

A linear system of three equations in four unknowns has

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

The row echelon form of the augmented matrix for a linear system is:

$[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.$

Example: Rank of a Matrix

Example:

1. Find the ranks of the following matrices and state the number of solutions.

a.) $\left[\begin{array}{l} 1& 0 \\ 2& 1 \\ 0 & 0 \\ \end{array}\right.\left|\begin{array}{l} 1\\2\\0\end{array}\right]$

The row echelon form of the augmented matrix for a linear system is:

$[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.$

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

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

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

Rank of a Matrix

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Rank of a Matrix

Full rank

Types of Solutions

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Example: Rank of a Matrix

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Extra Practice

Example: Invertible Matrix Theorem

Example:

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

Rank of a Matrix

Rank of a Matrix

Full rank

Types of Solutions

Example: Rank of a Matrix

Extra Practice

Example: Invertible Matrix Theorem

Example:

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