Mixing
Finding values of x for linear dependence
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Find all real numbers $x$ for which the matrices are linearly dependent
$$M_1=begin{bmatrix} 1&1\x&0 end{bmatrix}quad M_2= begin{bmatrix} 1&-x\x-1&3 end{bmatrix}$$
$$M_3= begin{bmatrix} 0&-2\1&x end{bmatrix}quad M_4= begin{bmatrix} 1&-3\-1&2x end{bmatrix}$$
Note: As usual, it will be useful if you start your answer by stating what condition you have to verify. Be careful not to divide by zero in your calculation.
To start this question I gave the condition of Linear dependence and set up a gaussian elimination with the last columns as 0's but this is as far as I can go as I have never worked with a question of this sort, so I need help on how to conduct the gaussian on this question
matrices gaussian-elimination
edited yesterday
Lorenzo B.
1,6222419
asked 2 days ago
connttttt
12
add a comment |
up vote
0
down vote
favorite
Find all real numbers $x$ for which the matrices are linearly dependent
$$M_1=begin{bmatrix} 1&1\x&0 end{bmatrix}quad M_2= begin{bmatrix} 1&-x\x-1&3 end{bmatrix}$$
$$M_3= begin{bmatrix} 0&-2\1&x end{bmatrix}quad M_4= begin{bmatrix} 1&-3\-1&2x end{bmatrix}$$
Note: As usual, it will be useful if you start your answer by stating what condition you have to verify. Be careful not to divide by zero in your calculation.
To start this question I gave the condition of Linear dependence and set up a gaussian elimination with the last columns as 0's but this is as far as I can go as I have never worked with a question of this sort, so I need help on how to conduct the gaussian on this question
matrices gaussian-elimination
edited yesterday
Lorenzo B.
1,6222419
asked 2 days ago
connttttt
12
Would you mind using MathJax to format your matrices? I’m not sure I understand what your matrices look like.
– Jack Moody
2 days ago
1
I have edited it
– connttttt
2 days ago
Are you trying to prove that the four matrices are linearly independent, or that each individual matrix has linearly-independent rows/columns?
– amd
yesterday
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Find all real numbers $x$ for which the matrices are linearly dependent
$$M_1=begin{bmatrix} 1&1\x&0 end{bmatrix}quad M_2= begin{bmatrix} 1&-x\x-1&3 end{bmatrix}$$
$$M_3= begin{bmatrix} 0&-2\1&x end{bmatrix}quad M_4= begin{bmatrix} 1&-3\-1&2x end{bmatrix}$$
Note: As usual, it will be useful if you start your answer by stating what condition you have to verify. Be careful not to divide by zero in your calculation.
To start this question I gave the condition of Linear dependence and set up a gaussian elimination with the last columns as 0's but this is as far as I can go as I have never worked with a question of this sort, so I need help on how to conduct the gaussian on this question
matrices gaussian-elimination
edited yesterday
Lorenzo B.
1,6222419
asked 2 days ago
connttttt
12
Find all real numbers $x$ for which the matrices are linearly dependent
$$M_1=begin{bmatrix} 1&1\x&0 end{bmatrix}quad M_2= begin{bmatrix} 1&-x\x-1&3 end{bmatrix}$$
$$M_3= begin{bmatrix} 0&-2\1&x end{bmatrix}quad M_4= begin{bmatrix} 1&-3\-1&2x end{bmatrix}$$
Note: As usual, it will be useful if you start your answer by stating what condition you have to verify. Be careful not to divide by zero in your calculation.
To start this question I gave the condition of Linear dependence and set up a gaussian elimination with the last columns as 0's but this is as far as I can go as I have never worked with a question of this sort, so I need help on how to conduct the gaussian on this question
matrices gaussian-elimination
matrices gaussian-elimination
edited yesterday
Lorenzo B.
1,6222419
asked 2 days ago
connttttt
12
edited yesterday
Lorenzo B.
1,6222419
asked 2 days ago
connttttt
12
edited yesterday
Lorenzo B.
1,6222419
edited yesterday
Lorenzo B.
1,6222419
edited yesterday
Lorenzo B.
1,6222419
1,6222419
asked 2 days ago
connttttt
12
asked 2 days ago
connttttt
12
asked 2 days ago
connttttt
12
12
Would you mind using MathJax to format your matrices? I’m not sure I understand what your matrices look like.
– Jack Moody
2 days ago
1
I have edited it
– connttttt
2 days ago
Are you trying to prove that the four matrices are linearly independent, or that each individual matrix has linearly-independent rows/columns?
– amd
yesterday
add a comment |
Would you mind using MathJax to format your matrices? I’m not sure I understand what your matrices look like.
– Jack Moody
2 days ago
1
I have edited it
– connttttt
2 days ago
Are you trying to prove that the four matrices are linearly independent, or that each individual matrix has linearly-independent rows/columns?
– amd
yesterday
Would you mind using MathJax to format your matrices? I’m not sure I understand what your matrices look like.
– Jack Moody
2 days ago
Would you mind using MathJax to format your matrices? I’m not sure I understand what your matrices look like.
– Jack Moody
2 days ago
1
1
I have edited it
– connttttt
2 days ago
I have edited it
– connttttt
2 days ago
Are you trying to prove that the four matrices are linearly independent, or that each individual matrix has linearly-independent rows/columns?
– amd
yesterday
Are you trying to prove that the four matrices are linearly independent, or that each individual matrix has linearly-independent rows/columns?
– amd
yesterday
add a comment |
1 Answer
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votes
up vote
1
down vote
You want to find where the determinant is zero. The determinant of matrix $A$, where $A$ is given by
$$A = begin{pmatrix}a & b\ c & d end{pmatrix}$$
The determinant of $A$ is $ad-bc$. So you want to solve $ad-bc = 0$.
For example, consider $M_{3}. The determinant is given by
$$text{det}(M_{3}) = begin{vmatrix}
0 & -2\
1 & x
end{vmatrix} = 0 cdot x + 1 cdot -2 =-2.$$
Since this determinant can never be zero, you have no values of $x$ such that the rows are linearly dependent.
Alternatively, you can find which scalar multiples work such that multiplying the first row by the scalar gives the second row.
answered 2 days ago
Jack Moody
16611
I am unclear as how the determinant would work for this example? Can you show me?
– connttttt
2 days ago
@connttttt see the updated answer
– Jack Moody
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
You want to find where the determinant is zero. The determinant of matrix $A$, where $A$ is given by
$$A = begin{pmatrix}a & b\ c & d end{pmatrix}$$
The determinant of $A$ is $ad-bc$. So you want to solve $ad-bc = 0$.
For example, consider $M_{3}. The determinant is given by
$$text{det}(M_{3}) = begin{vmatrix}
0 & -2\
1 & x
end{vmatrix} = 0 cdot x + 1 cdot -2 =-2.$$
Since this determinant can never be zero, you have no values of $x$ such that the rows are linearly dependent.
Alternatively, you can find which scalar multiples work such that multiplying the first row by the scalar gives the second row.
answered 2 days ago
Jack Moody
16611
I am unclear as how the determinant would work for this example? Can you show me?
– connttttt
2 days ago
@connttttt see the updated answer
– Jack Moody
2 days ago
add a comment |
up vote
1
down vote
You want to find where the determinant is zero. The determinant of matrix $A$, where $A$ is given by
$$A = begin{pmatrix}a & b\ c & d end{pmatrix}$$
The determinant of $A$ is $ad-bc$. So you want to solve $ad-bc = 0$.
For example, consider $M_{3}. The determinant is given by
$$text{det}(M_{3}) = begin{vmatrix}
0 & -2\
1 & x
end{vmatrix} = 0 cdot x + 1 cdot -2 =-2.$$
Since this determinant can never be zero, you have no values of $x$ such that the rows are linearly dependent.
Alternatively, you can find which scalar multiples work such that multiplying the first row by the scalar gives the second row.
answered 2 days ago
Jack Moody
16611
I am unclear as how the determinant would work for this example? Can you show me?
– connttttt
2 days ago
@connttttt see the updated answer
– Jack Moody
2 days ago
add a comment |
up vote
1
down vote
up vote
1
down vote
You want to find where the determinant is zero. The determinant of matrix $A$, where $A$ is given by
$$A = begin{pmatrix}a & b\ c & d end{pmatrix}$$
The determinant of $A$ is $ad-bc$. So you want to solve $ad-bc = 0$.
For example, consider $M_{3}. The determinant is given by
$$text{det}(M_{3}) = begin{vmatrix}
0 & -2\
1 & x
end{vmatrix} = 0 cdot x + 1 cdot -2 =-2.$$
Since this determinant can never be zero, you have no values of $x$ such that the rows are linearly dependent.
Alternatively, you can find which scalar multiples work such that multiplying the first row by the scalar gives the second row.
answered 2 days ago
Jack Moody
16611
You want to find where the determinant is zero. The determinant of matrix $A$, where $A$ is given by
$$A = begin{pmatrix}a & b\ c & d end{pmatrix}$$
The determinant of $A$ is $ad-bc$. So you want to solve $ad-bc = 0$.
For example, consider $M_{3}. The determinant is given by
$$text{det}(M_{3}) = begin{vmatrix}
0 & -2\
1 & x
end{vmatrix} = 0 cdot x + 1 cdot -2 =-2.$$
Since this determinant can never be zero, you have no values of $x$ such that the rows are linearly dependent.
Alternatively, you can find which scalar multiples work such that multiplying the first row by the scalar gives the second row.
answered 2 days ago
Jack Moody
16611
edited 2 days ago
answered 2 days ago
Jack Moody
16611
answered 2 days ago
Jack Moody
16611
answered 2 days ago
Jack Moody
16611
16611
I am unclear as how the determinant would work for this example? Can you show me?
– connttttt
2 days ago
@connttttt see the updated answer
– Jack Moody
2 days ago
add a comment |
I am unclear as how the determinant would work for this example? Can you show me?
– connttttt
2 days ago
@connttttt see the updated answer
– Jack Moody
2 days ago
I am unclear as how the determinant would work for this example? Can you show me?
– connttttt
2 days ago
I am unclear as how the determinant would work for this example? Can you show me?
– connttttt
2 days ago
@connttttt see the updated answer
– Jack Moody
2 days ago
@connttttt see the updated answer
– Jack Moody
2 days ago
add a comment |
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Would you mind using MathJax to format your matrices? I’m not sure I understand what your matrices look like.
– Jack Moody
2 days ago
1
I have edited it
– connttttt
2 days ago
Are you trying to prove that the four matrices are linearly independent, or that each individual matrix has linearly-independent rows/columns?
– amd
yesterday