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Determinant Multiple

Getting the determinant value of the original matrix from its upper triangular matrixWhy is the determinant invariant under row and column operations?Finding determinants using both reduction and cofactor expansionDerivation of the $2times 2$ determinantHow do I know if a matrix is irreducible?What is the simplest way to solve determinant of a $n times n$ matrix by upper and lower triangular matrices?Finding the determinant of a skew-symmetric matrix $K$Finding the determinant using row operations.Use elimination to find the determinantSolve the following nxn determinant by reducing it to a upper/lower triangular determinant

0

$begingroup$

I keep getting a multiple of the actual value of the determinant of matrices after performing row operations to reduce it to an upper triangular matrix. Any ideas why this might happen?For an easy 3x3 for example

matrices determinant

share|cite|improve this question

edited Mar 6 '18 at 14:43

user1726937

asked Mar 6 '18 at 11:52

user1726937user1726937

11

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to stackexchange. Please edit your question to show us just what you did. Then we might be able to help. Use mathjax: math.meta.stackexchange.com/questions/5020/…
  $endgroup$
  – Ethan Bolker
  Mar 6 '18 at 11:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It would be good if you include an example for a $2times 2$-matrix.
  $endgroup$
  – Dietrich Burde
  Mar 6 '18 at 12:22
  

  
  
  
  

add a comment | 

0

$begingroup$

I keep getting a multiple of the actual value of the determinant of matrices after performing row operations to reduce it to an upper triangular matrix. Any ideas why this might happen?For an easy 3x3 for example

matrices determinant

share|cite|improve this question

edited Mar 6 '18 at 14:43

user1726937

asked Mar 6 '18 at 11:52

user1726937user1726937

11

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to stackexchange. Please edit your question to show us just what you did. Then we might be able to help. Use mathjax: math.meta.stackexchange.com/questions/5020/…
  $endgroup$
  – Ethan Bolker
  Mar 6 '18 at 11:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It would be good if you include an example for a $2times 2$-matrix.
  $endgroup$
  – Dietrich Burde
  Mar 6 '18 at 12:22
  

  
  
  
  

add a comment | 

$begingroup$

I keep getting a multiple of the actual value of the determinant of matrices after performing row operations to reduce it to an upper triangular matrix. Any ideas why this might happen?For an easy 3x3 for example

matrices determinant

share|cite|improve this question

edited Mar 6 '18 at 14:43

user1726937

asked Mar 6 '18 at 11:52

user1726937user1726937

11

$endgroup$

share|cite|improve this question

edited Mar 6 '18 at 14:43

user1726937

asked Mar 6 '18 at 11:52

user1726937user1726937

11

share|cite|improve this question

edited Mar 6 '18 at 14:43

user1726937

asked Mar 6 '18 at 11:52

user1726937user1726937

11

asked Mar 6 '18 at 11:52

user1726937user1726937

11

asked Mar 6 '18 at 11:52

user1726937user1726937

11

1 Answer
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oldest

votes

0

$begingroup$

Each and every Row/Column operations will affect the value of the determinant depends on the constant that we are multiplying for the row.

For example,

$R_1 -> R_1 - R_2$ will not affect the value of the determinant.

But, $R_1 -> 2R_1 - R_2$ will double the value of the determinant as we are multiplying the $R_1$ by 2.

Even the operation $R_1 -> R_2 - R_1$, will actually affect the determinant value as $-1 * Determinant$

So, At each steps whenever we are doing such operations, we should ensure that we have the Same Co-efficient with sign on both the side. If you want to do the operations by multiplying with some constants, then we should take the constant outside as a divisor.

For example,

$R_1 -> R_1 - R_2$ --> $Actual,Determinant = Determinant,,after,Row/Column,operation$

$R_1 -> R_2 - R_1$ --> $Actual,Determinant = fracDeterminant,,after,Row/Column,operation-1$

$R_2 -> -3R_2 - 5R_1$ --> $Actual,Determinant = fracDeterminant,,after,Row/Column,operation-3$

In Your attached example,

At first step you have done $C_1 -> 5C_2 - 6C_1$ --> $Actual,Determinant = (1/-6),(Det)$

At Second step you have done $R_3 -> R_1 + R_3$ --> $Actual,Determinant = (1/1),(Det)$

At Third step you have done $R_2 -> 2R_2 - R_3$ --> $Actual,Determinant = (1/-1),(Det)$

You got your determinant value as 162 after these operations.

So,
$Actual,Determinant = frac162(-6)(1)(-1),=27$

share|cite|improve this answer

answered Mar 20 at 19:36

Saravanakumar VSaravanakumar V

11

$endgroup$

add a comment | 

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0

$begingroup$

Each and every Row/Column operations will affect the value of the determinant depends on the constant that we are multiplying for the row.

For example,

$R_1 -> R_1 - R_2$ will not affect the value of the determinant.

But, $R_1 -> 2R_1 - R_2$ will double the value of the determinant as we are multiplying the $R_1$ by 2.

Even the operation $R_1 -> R_2 - R_1$, will actually affect the determinant value as $-1 * Determinant$

So, At each steps whenever we are doing such operations, we should ensure that we have the Same Co-efficient with sign on both the side. If you want to do the operations by multiplying with some constants, then we should take the constant outside as a divisor.

For example,

$R_1 -> R_1 - R_2$ --> $Actual,Determinant = Determinant,,after,Row/Column,operation$

$R_1 -> R_2 - R_1$ --> $Actual,Determinant = fracDeterminant,,after,Row/Column,operation-1$

$R_2 -> -3R_2 - 5R_1$ --> $Actual,Determinant = fracDeterminant,,after,Row/Column,operation-3$

In Your attached example,

At first step you have done $C_1 -> 5C_2 - 6C_1$ --> $Actual,Determinant = (1/-6),(Det)$

At Second step you have done $R_3 -> R_1 + R_3$ --> $Actual,Determinant = (1/1),(Det)$

At Third step you have done $R_2 -> 2R_2 - R_3$ --> $Actual,Determinant = (1/-1),(Det)$

You got your determinant value as 162 after these operations.

So,
$Actual,Determinant = frac162(-6)(1)(-1),=27$

share|cite|improve this answer

answered Mar 20 at 19:36

Saravanakumar VSaravanakumar V

11

$endgroup$

add a comment | 

0

$begingroup$

Each and every Row/Column operations will affect the value of the determinant depends on the constant that we are multiplying for the row.

For example,

$R_1 -> R_1 - R_2$ will not affect the value of the determinant.

But, $R_1 -> 2R_1 - R_2$ will double the value of the determinant as we are multiplying the $R_1$ by 2.

Even the operation $R_1 -> R_2 - R_1$, will actually affect the determinant value as $-1 * Determinant$

So, At each steps whenever we are doing such operations, we should ensure that we have the Same Co-efficient with sign on both the side. If you want to do the operations by multiplying with some constants, then we should take the constant outside as a divisor.

For example,

$R_1 -> R_1 - R_2$ --> $Actual,Determinant = Determinant,,after,Row/Column,operation$

$R_1 -> R_2 - R_1$ --> $Actual,Determinant = fracDeterminant,,after,Row/Column,operation-1$

$R_2 -> -3R_2 - 5R_1$ --> $Actual,Determinant = fracDeterminant,,after,Row/Column,operation-3$

In Your attached example,

At first step you have done $C_1 -> 5C_2 - 6C_1$ --> $Actual,Determinant = (1/-6),(Det)$

At Second step you have done $R_3 -> R_1 + R_3$ --> $Actual,Determinant = (1/1),(Det)$

At Third step you have done $R_2 -> 2R_2 - R_3$ --> $Actual,Determinant = (1/-1),(Det)$

You got your determinant value as 162 after these operations.

So,
$Actual,Determinant = frac162(-6)(1)(-1),=27$

share|cite|improve this answer

answered Mar 20 at 19:36

Saravanakumar VSaravanakumar V

11

$endgroup$

add a comment | 

$begingroup$

Each and every Row/Column operations will affect the value of the determinant depends on the constant that we are multiplying for the row.

For example,

$R_1 -> R_1 - R_2$ will not affect the value of the determinant.

But, $R_1 -> 2R_1 - R_2$ will double the value of the determinant as we are multiplying the $R_1$ by 2.

Even the operation $R_1 -> R_2 - R_1$, will actually affect the determinant value as $-1 * Determinant$

So, At each steps whenever we are doing such operations, we should ensure that we have the Same Co-efficient with sign on both the side. If you want to do the operations by multiplying with some constants, then we should take the constant outside as a divisor.

For example,

$R_1 -> R_1 - R_2$ --> $Actual,Determinant = Determinant,,after,Row/Column,operation$

$R_1 -> R_2 - R_1$ --> $Actual,Determinant = fracDeterminant,,after,Row/Column,operation-1$

$R_2 -> -3R_2 - 5R_1$ --> $Actual,Determinant = fracDeterminant,,after,Row/Column,operation-3$

In Your attached example,

At first step you have done $C_1 -> 5C_2 - 6C_1$ --> $Actual,Determinant = (1/-6),(Det)$

At Second step you have done $R_3 -> R_1 + R_3$ --> $Actual,Determinant = (1/1),(Det)$

At Third step you have done $R_2 -> 2R_2 - R_3$ --> $Actual,Determinant = (1/-1),(Det)$

You got your determinant value as 162 after these operations.

So,
$Actual,Determinant = frac162(-6)(1)(-1),=27$

share|cite|improve this answer

answered Mar 20 at 19:36

Saravanakumar VSaravanakumar V

11

$endgroup$

share|cite|improve this answer

answered Mar 20 at 19:36

Saravanakumar VSaravanakumar V

11

answered Mar 20 at 19:36

Saravanakumar VSaravanakumar V

11

answered Mar 20 at 19:36

Saravanakumar VSaravanakumar V

11