Rank of a Matrix

4 Activities

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

A=[2300002−1]A=\left[\begin{array}{rrr|r}
2&3&0&0\\
0&0&2&-1
 \end{array}\right]A=[20​30​02​0−1​]

B=[000000]B=\left[\begin{array}{rr|r}
0&0&0\\
0&0&0
 \end{array}\right]B=[00​00​00​]

C=[23000001−1]C=\left[\begin{array}{rr|r}
2&3&0\\
0&0&0\\
0&1&-1
 \end{array}\right]C=​200​301​00−1​​

D=[100000100100]D=\left[\begin{array}{rrr|r}
1&0&0&0\\
0&0&1&0\\
0&1&0&0
 \end{array}\right]D=​100​001​010​000​​

E=[2300000−10121]E=\left[\begin{array}{rrr|r}
2&3&0&0\\
0&0&0&-1\\
0&1&2&1
 \end{array}\right]E=​200​301​002​0−11​​

I don't know

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 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

Related Topics

Rank of a Matrix

More Rank of a Matrix Questions:

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 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