The row echelon form of the augmented matrix for a linear system is: [A|b][{l 1…

Solving Linear Systems

6 Activities

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

Rank of a Matrix

4 Activities

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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 rank of matrix A
Does this linear system have a unique solution, infinity many solutions or no solution?
Find all solutions of homogeneous system.

I have answered this question

Find size of soln set to nonhom linear system

Find all solutions to the system of linear equations 5x+y+z=0x+2z=−12x+y−5z=1\begin{array}{rcl}
5x+y+z=0\\
x+2z=-1\\
2x+y-5z=1
\end{array}5x+y+z=0x+2z=−12x+y−5z=1​.

Given the augmented matrix  [1201010200k2−9k−3]\left[\begin{array}{rrr|r}
1 & 2 &0 &1  \\

0 & 1  &0&2\\
0&0&k^2-9&k-3
 \end{array}\right]​100​210​00k2−9​12k−3​​, answer the following questions.

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 44.4

Interpret the following matrix as the RREF of a matrix of coefficients augmented by the vector of constants and determine whether the systems  has a unique solution.
[100001000010] \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} ​100​010​001​000​​

 Find the all values of p and q such that the system has at least one solution.
[12401300p∣15q]\left[\begin{array}{l}
1&2&4\\
0&1&3\\
0&0&p\\
\end{array}\right.\left|\begin{array}{l}1
\\5\\q\end{array}\right]​100​210​43p​​15q​​

 Find the value of p such that there is a unique solution to this system of equations:
L:{         2x1+x3=4x1+3x2+px3=−10     x1−x2+3x3=5L:\begin{cases}
\ \ \ \ \ \ \ \ \ 2x_1+x_3=4\\
x_1+3x_2+px_3=-10\\
 \ \ \ \ \ x_1-x_2+3x_3=5
\end{cases}L:⎩⎨⎧​         2x1​+x3​=4x1​+3x2​+px3​=−10     x1​−x2​+3x3​=5​

a) For what values of c does the system have one solution?
{2𝑥+𝒄𝑦=0𝒄𝑥+𝑦=3\begin{cases}
2𝑥 + 𝒄𝑦 = 0\\
𝒄𝑥 + 𝑦 = 3
\end{cases}{2x+cy=0cx+y=3​
b) For what values of c does it have infinitely many solutions?

For what value of p does the following system of equations have no solution? What is the geometric interpretation of these equations?
x+3y=1x+3y=1x+3y=1
px−6y=−4px-6y=-4px−6y=−4

Practice Question: Solving a System of Linear Equations (Infinitely Many Solutions)

Practice Question: Solving a SLE
Solve the following system of linear equations: x+2y=4y+z=3z+w=1\begin{array}{rcl}
x+2y&=&4\\
y+z&=&3\\
z+w&=&1
\end{array}x+2yy+zz+w​===​431​.

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​​.​

Which of the following options is the general solution of the following system of linear equations?
x−y−z=2x+y+z=0\begin{array}{l}x-y-z=2\\x+y+z=0\end{array}x−y−z=2x+y+z=0​

Find the basic solutions of the following homogeneous system of linear equations:
x1−2x2+3x3+x4+2x5=0x1−x2+3x3−x4+3x5=0−x1+2x2−4x3+x4+x5=0\begin{array}{l}x_1 - 2x_2 + 3x_3 + x_4 + 2x_5 = 0\\x_1 - x_2 + 3x_3 - x_4 + 3x_5 = 0\\-x_1 + 2x_2 - 4x_3 + x_4 + x_5 = 0\end{array}x1​−2x2​+3x3​+x4​+2x5​=0x1​−x2​+3x3​−x4​+3x5​=0−x1​+2x2​−4x3​+x4​+x5​=0​

What can you say about the following systems of linear equations?
x−2y−3z=42x+2y−6z=−1\begin{array}{l}x-2y-3z=4\\2x+2y-6z=-1\end{array}x−2y−3z=42x+2y−6z=−1​

What can you say about the following systems of linear equations?
x−y+3z=52x−2y+6z=−10\begin{array}{l}x-y+3z=5\\2x-2y+6z=-10\end{array}x−y+3z=52x−2y+6z=−10​

Write the general solution to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l} x - y - 3z = -4\\ 2x - 3y + z = 0 \end{array}x−y−3z=−42x−3y+z=0​

Wize Concept Clarifier--Gaussian Jordan Elimination

Wize Concept Clarifier
Solve the system x+y−z=24y−4z=12x+3y−3z=−1\begin{array}{}
x+y-z=2\\
4y-4z=1\\
2x+3y-3z=-1
\end{array}x+y−z=24y−4z=12x+3y−3z=−1​.

Linear Systems - Mixing

You are the cook on a celebrity yacht, and you have run out of flour, sugar, and baking powder -
except for that contained in a large quantity of 3 different pancake mixes:
Mix A: 8 parts flour to 8 parts sugar to 1 part baking powder

Basics of Systems of Linear Equations

Practice: Solutions to Linear Systems
Determine which vectors are solutions to the following SLE [select all that apply]:
x1 − 5x2 +x3=2−x1 +5x2 −3x3= −6\begin{alignedat}{8}
x_1\ &-&\ 5x_2\ &+& x_3 &=& 2\\
-x_1\ &+& 5x_2\ &-& 3x_3 &=&\ -6
\end{alignedat}x1​ −x1​ ​−+​ 5x2​ 5x2​ ​+−​x3​3x3​​==​2 −6​

123 - FML #1 - 18.1W - e.g. 20

Consider the following system of linear equations: 
		{x−2y+4z=22x+y−2z=−13x−y+2z=12x+6y−12z=−6\bcb{

\begin{cases}
					x - 2y + 4z = 2& \\
					2x + y - 2z = -1 &\\
					3x - y + 2z = 1 &\\
					2x + 6y - 12z = -6& \end{cases}}⎩⎨⎧​x−2y+4z=22x+y−2z=−13x−y+2z=12x+6y−12z=−6​​				
a) Construct the augmented matrix A\bcb{A}A corresponding to this system.

Solve linear systems

Solve the following systems of linear equations:
a) x+2y=−3−x+3y=1\begin{array}{r}x+2y=-3\\-x+3y=1\end{array}x+2y=−3−x+3y=1​
b) x+y−2z=1−4x+3y+2z=1−3x+11x−6z=0\begin{array}{r}
x+y-2z=1\\
-4x+3y+2z=1\\
-3x+11x-6z=0
\end{array}x+y−2z=1−4x+3y+2z=1−3x+11x−6z=0​

123 - FML #1 - 18.1W - e.g. 21.1

Solve the following system of linear equations:
 {2x+y+z+3w=13x−2z+2w=−15x+y−z+5w=07x+2y+8w=1\bcb{ \begin{cases}
										2x + y + z + 3w = 1 \\
										3x - 2z + 2w = -1 \\
										5x + y - z + 5w = 0 \\
										7x + 2y + 8w = 1
										\end{cases} }⎩⎨⎧​2x+y+z+3w=13x−2z+2w=−15x+y−z+5w=07x+2y+8w=1​ 

Solve the following linear system of equations by Gaussian Elimination.
2x−y+5z=14x+y+z=4−x−3y+3z=−2\begin{array}{c}
2x - y + 5z = 14\\
x + y + z = 4\\
-x - 3y + 3z = - 2
\end{array}2x−y+5z=14x+y+z=4−x−3y+3z=−2​

Solve the following linear system of equations.
3x+2y−2z=185x+2y−z=27x+y+z=9\begin{array}{c}
3x + 2y - 2z = 18\\
5x + 2y - z =27\\
x + y + z = 9
\end{array}3x+2y−2z=185x+2y−z=27x+y+z=9​

Find soln to nonhom linear system

Find all solutions to the following system of linear equations[10210131]\left[\begin{array}{rrr|r}
1&0&2&1\\
0&1&3&1
\end{array}\right][10​01​23​11​].

Balancing Chemical Equations

The chemical equation for the combustion of octane (the primary component of gasoline) is:
C8H18+O2=H2O+CO2C_8H_{18}+O_2 = H_2O+CO_2C8​H18​+O2​=H2​O+CO2​
Balance this equation using Gauss-Jordan elimination

Find all possible solutions to the following linear system of equations.
x+2y−z=9x−y+2z=3\begin{array}{c}
x + 2y - z = 9\\
x - y + 2z = 3
\end{array}x+2y−z=9x−y+2z=3​

Solving Linear Systems

Practice: Solving a Linear System
Solve the following system of linear equations. What type of solution is it?
3x1+6x2=12x1+4x2=2x2+x3=3\begin{aligned}
3x_1 + 6x_2&= 1\\
2x_1 + 4x_2 &=2\\
x_2 + x_3 &= 3  
\end{aligned}3x1​+6x2​2x1​+4x2​x2​+x3​​=1=2=3​

Solving Linear Systems

Practice: Solving a Homogeneous Linear System
Solve the following homogeneous system of linear equations. What type of solution is it?
2x1−x2+2x4=02x2+4x3−x4=0\begin{aligned}
2x_1-x_2+2x_4 &= 0\\
2x_2 + 4x_3 − x_4 &= 0
\end{aligned}2x1​−x2​+2x4​2x2​+4x3​−x4​​=0=0​

Solving Linear Systems

Practice: Geometric Interpretation of a Linear System
Consider the following augmented matrix in RREF:
[1−100−540010−2200014−3]\left[\begin{array}{rrrrr|r}
1&-1&0&0&-5&4\\[0.5em]
0&0&1&0&-2&2\\[0.5em]
0&0&0&1&4&-3
\end{array}\right]​100​−100​010​001​−5−24​42−3​​

 Find I0 in the following circuit.

Solving Linear Systems

Example: Solving a Linear System (Unique Solution)
Solve the system of linear equations:
2x+2y−2z=2x+2z−7=03y=−7−2z\begin{array}{ccccccl}
2x&+&2y&-&2z&=&2\\
x&+&2z&-&7&=&0\\
&&&&3y &=&-7-2z
\end{array}2xx​++​2y2z​−−​2z73y​===​20−7−2z​

Solving Linear Systems

Example: Solving a Linear System (Infinitely Many Solutions)
Solve the following system of linear equations:
x+2y=4y+z=3z+w=1\begin{array}{c}
x+2y&=&4\\
y+z&=&3\\
z+w&=&1
\end{array}x+2yy+zz+w​===​431​

19.4F_Final_Builder_Ch_2.13_Algebra_SLE_$\tkco{eg22}$_$\key{Final}$_Builder_$\tkcth{2.13.}\tkcf{22}$_

Given the system of equations:
{2x1−4x2=−1−4x1+8x2=2
            \left\{
            \begin{array}{ll}
              2x_1 - 4x_2 & = \quad -1 \\
              -4x_1 + 8x_2 & = \quad 2
            \end{array}
            \right.
        {2x1​−4x2​−4x1​+8x2​​=−1=2​
Show that x⃗g=x⃗p+tx⃗h\vx_g = \vx_p + t \vx_hxg​=xp​+txh​, where x⃗p=[−120]\vx_p = \colvec{-\frac12}{0}xp​=[−21​0​] and x⃗h=[21]\vx_h = \colvec{2}{1}xh​=[21​] is a solution to the system when t=1,2,3t = 1, 2, 3t=1,2,3, i.e. show that the vectors:

Find (if possible) conditions on aaa and bbb so that the following system has
(a) no solution
(b) one solution

Which of the following options is the general solution of the following system of linear equations?
x−2y+z=3x−y−z=0\begin{array}{l}x-2y+z=3\\x-y-z=0\end{array}x−2y+z=3x−y−z=0​

What can you say about the following systems of linear equations?
x+y−2z=22x+2y−6z=3\begin{array}{l}x+y-2z=2\\2x+2y-6z=3\end{array}x+y−2z=22x+2y−6z=3​

What can you say about the following systems of linear equations?
x+y−2z=−32x+2y−4z=−6\begin{array}{l}x+y-2z=-3\\2x+2y-4z=-6\end{array}x+y−2z=−32x+2y−4z=−6​

Interpretation of Linear Systems

Which of the following describes the intersection between the following 3 hyperplanes in R4\reals^4R4?
2x1+3x3+x4=1x2−x4=0x2=−3\begin{array}{rcl}
2x_1+3x_3+x_4&=&1\\
x_2-x_4&=&0\\
x_2&=&-3
\end{array}2x1​+3x3​+x4​x2​−x4​x2​​===​10−3​

Matrix Multiplication

e.g. Find a 3 × 3 matrix M such that 
 ⁣M[   0−2   0]=[−1   1−1] ⁣, ⁣M[001]=[   0−2   3] ⁣,and M[1300]=[−2   2−1]\!M\begin{bmatrix}
\ \ \ 0\\-2\\\ \ \ 0
\end{bmatrix}=
\begin{bmatrix}
-1\\\ \ \ 1\\-1
\end{bmatrix}\!, \hspace{0.7cm}

\!M\begin{bmatrix}
0\\0\\1
\end{bmatrix}=
\begin{bmatrix}
\ \ \ 0\\-2\\\ \ \ 3
\end{bmatrix}\!, \hspace{0.7cm} \text{\small{and}}\ 

M\begin{bmatrix}
\frac{1}{3}\\0\\0
\end{bmatrix}=
\begin{bmatrix}
-2\\\ \ \ 2\\-1
\end{bmatrix}M​   0−2   0​​=​−1   1−1​​,M​001​​=​   0−2   3​​,and M​31​00​​=​−2   2−1​​

System of Linear Equations

Find (if possible) conditions on aaa and bbb so that the following system has
(a) no solution
(b) one solution

Homogeneous System of Equations

Find basic solutions of a homogeneous system of linear equations with reduced row echelon form of:
[1203000150]\left[\begin{array}{cccc|c}1 & 2 & 0 & 3 & 0 \\0 & 0 & 1 & 5 & 0\end{array}\right][10​20​01​35​00​]

Practice Question: Solving a System of Linear Equations (Unique Solution)

Practice Question: Solving a SLE--Unique Solution
Solve the following systems of linear equations.
   𝑥−𝑧=5𝑥−𝑧=5x−z=5

Find size of soln set to nonhom linear system

Find all solutions to the system of linear equations 5x+y+z=0x+2z=−12x+y−5z=1\begin{array}{rcl}
5x+y+z=0\\
x+2z=-1\\
2x+y-5z=1
\end{array}5x+y+z=0x+2z=−12x+y−5z=1​.

What can you say about the following systems of linear equations?
x+y−2z=22x+2y−6z=3\begin{array}{l}x+y-2z=2\\2x+2y-6z=3\end{array}x+y−2z=22x+2y−6z=3​

Write the general solution (all solutions) to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l}x-y-3z=-4\\2x-3y+z=0\end{array}x−y−3z=−42x−3y+z=0​

Find (if possible) conditions on aaa and bbb so that the following system has
(a) no solution
(b) one solution

What can you say about the following systems of linear equations?
x+y−2z=−32x+2y−4z=−6\begin{array}{l}x+y-2z=-3\\2x+2y-4z=-6\end{array}x+y−2z=−32x+2y−4z=−6​

Which of the following options is the general solution of the following system of linear equations?
x−2y+z=3x−y−z=0\begin{array}{l}x-2y+z=3\\x-y-z=0\end{array}x−2y+z=3x−y−z=0​

Find all solutions to the system of linear equations 5x+y+z=0x+2z=−12x+y−5z=1\begin{array}{c}
5x+y+z=0\\
x+2z=-1\\
2x+y-5z=1
\end{array}5x+y+z=0x+2z=−12x+y−5z=1​.

Practice Question: Solving a System of Linear Equations (Infinitely Many Solutions)

Practice Question: Solving a SLE
Solve the following system of linear equations: x+2y=4y+z=3z+w=1\begin{array}{rcl}
x+2y&=&4\\
y+z&=&3\\
z+w&=&1
\end{array}x+2yy+zz+w​===​431​.

Solve the homogeneous linear system

When the right hand side of every equation in a linear system is 0, we say that the system is homogeneous 
Find the general solution of the following homogeneous linear system:
{3x+2y=0−6x+4y=0\left\{\begin{array}{rcl}
3x+2y&=&0\\
-6x+4y&=&0
\end{array}
\right.{3x+2y−6x+4y​==​00​

Solving a system of linear equations

Solve the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr}
-2x_1+3x_2+13x_3&=&5\\
-x_1+2x_2+5x_3&=&1\\
4x_1-14x_2-2x_3&=&14\\
\end{array}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​

Solving a system of linear equations

Solve the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr}
-2x_1+3x_2+13x_3&=&5\\
-x_1+2x_2+5x_3&=&1\\
4x_1-14x_2-2x_3&=&14\\
\end{array}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​

Solving a system of linear equations

Solve the following system of linear equations:
−2x1−4x2−6x3−19x4=−153x1+10x2−30x3+9x4=24x1+3x2−6x3+4x4=87x1+17x2+45x4=55\begin{array}{rcr}
-2x_1-4x_2-6x_3-19x_4&=&-15\\
3x_1+10x_2-30x_3+9x_4&=&24\\
x_1+3x_2-6x_3+4x_4&=&8\\
7x_1+17x_2+45x_4&=&55
\end{array}−2x1​−4x2​−6x3​−19x4​3x1​+10x2​−30x3​+9x4​x1​+3x2​−6x3​+4x4​7x1​+17x2​+45x4​​====​−1524855​
where we wish to solve for the quadruple (x1,x2,x3,x4)(x_1,x_2,x_3,x_4)(x1​,x2​,x3​,x4​) of real numbers that satisfy this system

Practice: Solutions to a System of Equations

Practice: Solutions to a System of Equations
Given the system of linear equations x+y=0x−cy=1\begin{array}{c}
x+y=0\\
x-cy=1
\end{array}x+y=0x−cy=1​, find the values of ccc such that,

Solving a system of linear equations

Solve the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr}
-2x_1+3x_2+13x_3&=&5\\
-x_1+2x_2+5x_3&=&1\\
4x_1-14x_2-2x_3&=&14\\
\end{array}−2x1​+3x2​+13x3​−x1​+2x2​+5x3​4x1​−14x2​−2x3​​===​5114​
where we wish to solve for the triple (x1,x2,x3)(x_1,x_2,x_3)(x1​,x2​,x3​) of real numbers that satisfy this system

Practice: Solving a System of Equations

Practice: Solving a System of Equations
Solve the system of equations 2x+2y−2z=2x+2z−7=03y=−7−2z\begin{array}{rcl}
2x+2y-2z&=&2\\
x+2z-7&=&0\\
3y&=&-7-2z
\end{array}2x+2y−2zx+2z−73y​===​20−7−2z​

Linear algebra

For which values of aaaand bbbdoes the system x+3y=2b3x−3ay=a\begin{array}{l}x+3y = 2b\\3x-3ay = a\end{array}x+3y=2b3x−3ay=a​have infinitely many solutions?

For which values of a and b does the system x+3y=2b3x−3ay=a\begin{array}{l}x+3y = 2b\\3x-3ay = a\end{array}x+3y=2b3x−3ay=a​have infinitely many solutions?

For which values of a and b does the system 2x−4y=a3x−ay=3b\begin{array}{l}2x-4y = a\\3x-ay = 3b\end{array}2x−4y=a3x−ay=3b​have no solutions?

Write the general solution to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l} x - y - 3z = -4\\ 2x - 3y + z = 0 \end{array}x−y−3z=−42x−3y+z=0​

System of Linear Equations

Write the general solution to the system of linear equations
x−y−3z=−42x−3y+z=0\begin{array}{l} x - y - 3z = -4\\ 2x - 3y + z = 0 \end{array}x−y−3z=−42x−3y+z=0​

Solve the following linear system of equations.
2x+3y−z=7x−2y+5z=27y−11z=6\begin{array}{c}
2x + 3y - z = 7\\
x - 2y + 5z = 2\\
7y - 11z = 6
\end{array}2x+3y−z=7x−2y+5z=27y−11z=6​

Solving Linear Systems

Solve the following homogeneous system of linear equations. What type of solution is it?
2x1−x2+2x4=02x2+4x3−x4=0\begin{aligned}
2x_1-x_2+2x_4 &= 0\\
2x_2 + 4x_3 − x_4 &= 0
\end{aligned}2x1​−x2​+2x4​2x2​+4x3​−x4​​=0=0​

Solving Linear Systems

Solve the following system of linear equations. What type of solution is it?
3x1+6x2=12x1+4x2=2x2+x3=3\begin{aligned}
3x_1 + 6x_2&= 1\\
2x_1 + 4x_2 &=2\\
x_2 + x_3 &= 3  
\end{aligned}3x1​+6x2​2x1​+4x2​x2​+x3​​=1=2=3​

Interpretation of Linear Systems

Which of the following describes the intersection between the following 3 hyperplanes in R4\reals^4R4?
2x1+3x3+x4=1x2−x4=0x2=−3\begin{array}{rcl}
2x_1+3x_3+x_4&=&1\\
x_2-x_4&=&0\\
x_2&=&-3
\end{array}2x1​+3x3​+x4​x2​−x4​x2​​===​10−3​

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

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)

The row echelon form of the augmented matrix for a linear system is: [A|b][{l 1…

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

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