When dealing with upper and lower triangular matrices, the idea is to generate zeros in specific positions to form either an upper triangular matrix or a lower triangular matrix. If you're unable to easily create zeros directly, you might need to use row operations such as multiplication and addition to achieve this. Here are some methods to transform a matrix into upper or lower triangular form when you can't easily create zeros: Gaussian Elimination: Use row operations like multiplying rows by a scalar and adding multiples of one row to another to create zeros in desired positions. Continue this process until the matrix takes on the desired upper or lower triangular form. Elementary Matrices: Perform row operations using elementary matrices. An elementary matrix represents an elementary row operation (such as swapping rows, multiplying a row by a scalar, or adding a multiple of one row to another). Multiply your original matrix by these elementary matrices to transform it into the desired form. LU Decomposition: Factorize a matrix into a product of a lower triangular matrix (L) and an upper triangular matrix (U). If you have a matrix A, such that A = LU, where L is lower triangular and U is upper triangular, you can use this factorization to solve systems of linear equations or represent the matrix in upper/lower triangular form. When performing row operations or using elementary matrices, the key is to maintain the equality of the matrix transformations. If you multiply the original matrix by a certain matrix on the left, ensure you perform the same operation on the right to maintain equality. Remember, matrices can be transformed in various ways using elementary row operations, and the choice of which operation to use at a particular step depends on the specific elements and structure of the matrix you're working with.
Just found your video and it was a good explanation on the conversion. I tried doing it myself by pausing the video and what I got for the determinant is -30. I didn’t perform all the same operations that you did but I was still able to get the matrix into upper triangular form. My confusion is coming from the fact that we aren’t getting back the same results. I did the same operation as you for R3 and R4 in the same exact order. I then went on to get -3 in column2 to become 0 by the operation “R3= 2*R3 + 3*R2” and the 4 in column 3 to 0 by the operation “R4=5*R4 + 4*R3”. What I ended up with is multiplying 1*2*-5*3 = -30 as the determinant. Can you please help by explaining to me where I went wrong because I need help.
Let's walk through the steps to ensure we understand where the discrepancy might be coming from in calculating the determinant. Given the steps you've mentioned, it looks like you're transforming a matrix into upper triangular form to find the determinant. The determinant of a matrix can be calculated by transforming it to upper triangular form and then multiplying the diagonal elements, taking into account any row swaps and scaling factors used during row operations. Let's denote the matrix as \( A \). Here's the general process: 1. **Row operations to achieve upper triangular form:** - Only row swaps affect the determinant by a factor of \(-1\) each time. - Multiplying a row by a scalar \( k \) scales the determinant by \( k \). - Adding or subtracting multiples of one row to another does not change the determinant. 2. **Calculation of the determinant:** - After converting \( A \) to an upper triangular matrix \( U \), the determinant of \( A \) is the product of the diagonal elements of \( U \), adjusted by the factors from row swaps and row scalings. Let's verify the operations you performed. Assume your original matrix \( A \) is: \[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix} \] Here are the steps you mentioned: 1. **Operations on \( R3 \) and \( R4 \):** - You applied \( R3 = 2*R3 + 3*R2 \). This operation does not change the determinant. - You applied \( R4 = 5*R4 + 4*R3 \). This operation does not change the determinant. 2. **Achieving zeros in column 2 and column 3:** - After these operations, you should have an upper triangular matrix. Let's denote it as \( U \). 3. **Calculating the determinant:** - The determinant of the upper triangular matrix \( U \) is the product of its diagonal elements. Let's assume you ended up with the following upper triangular matrix \( U \) after all operations (this is an illustration, as I don't have the exact matrix): \[ U = \begin{pmatrix} u_{11} & u_{12} & u_{13} & u_{14} \\ 0 & u_{22} & u_{23} & u_{24} \\ 0 & 0 & u_{33} & u_{34} \\ 0 & 0 & 0 & u_{44} \end{pmatrix} \] The determinant of \( U \) is: \[ \text{det}(U) = u_{11} \cdot u_{22} \cdot u_{33} \cdot u_{44} \] ### Check for Row Swaps and Scaling Factors: - If there were any row swaps, each swap would multiply the determinant by \(-1\). - If there were any row scalings by a factor \( k \), the determinant would be scaled by \( \frac{1}{k} \). Given your determinant calculation of \( -30 \), you mentioned multiplying factors \( 1 \times 2 \times -5 \times 3 \): - Ensure that these factors correspond to the diagonal elements of the final upper triangular matrix. - Verify if any row swaps occurred (which would introduce a \(-1\) factor for each swap). - Check if there were any row scalings, which would adjust the determinant accordingly. Without the exact initial matrix and the intermediate steps, it's challenging to pinpoint the exact error. However, common mistakes might include: - Missing a row swap or a row scaling. - Incorrect application of row operations leading to incorrect upper triangular form. - Arithmetic errors during row operations.
No, if you follow different methods to compute the determinant of the same matrix, the result must always be the same. The determinant is a unique scalar value associated with a square matrix, and its value does not depend on the method used, as long as all the steps follow valid mathematical rules. In your example, if the determinant of matrix 𝐴 A is calculated to be 3, then any correct method of calculating it must also result in 3. If you use a different method and obtain 72, then an error likely occurred in your calculations. The steps for calculating the determinant (such as row reduction, cofactor expansion, or other techniques) must all lead to the same result. If you get different results using different methods, it's important to review your calculations to ensure all steps were done correctly.
There is no strict rule that the diagonal must be all 1s for determinant calculation. That’s more relevant for achieving reduced row echelon form. The determinant is invariant under row addition/subtraction but changes sign when rows are swapped and scales if rows are multiplied by constants. Pivoting and Gaussian elimination focus on numerical stability rather than affecting the determinant directly, except for the mentioned row swaps or scaling.
When finding the determinant of a matrix using row operations, the determinant is multiplied by a scalar factor whenever a row operation is performed. However, it is important to remember that the determinant is a measure of the "scale factor" of the linear transformation represented by the matrix. Multiplying a row of a matrix by a scalar factor changes the scale of the linear transformation represented by the matrix. If you multiply a row by 3, for example, you are effectively scaling the linear transformation by a factor of 3 along that particular row. However, if you multiply the determinant by 3, you are effectively scaling the entire linear transformation by a factor of 3, which is not the same as just scaling one row of the matrix. This will result in an incorrect value for the determinant, since it is not measuring the scale factor of the original linear transformation. Therefore, you should only multiply the determinant by a scalar factor that corresponds to the row operation being performed, not a scalar factor that applies to the entire matrix.
@@yesdev I don't think it answers his question, you multiplied a row by 3 in your calculation, why didn't you multiply it by the determinant of the resulting matrix? More explanations please✨🤲
is Gaussian elimination, which is also known as row reduction or row echelon form. these matrices can be obtained through various methods such as LU decomposition, Cholesky decomposition, or simply by performing row operations on the original matrix. However, there isn't a specific method named after "upper and lower triangular matrices" themselves. They are usually produced as intermediate steps or results of other matrix operations
In the context of performing row operations on matrices, it's important to consider the properties and structure of upper and lower triangular matrices. An upper triangular matrix is a square matrix in which all entries below the main diagonal (from the top left to the bottom right) are zero. A lower triangular matrix is a square matrix in which all entries above the main diagonal are zero. When you perform a row operation like R3' = R3 + (3/2)R2, you are adding a multiple of one row to another row. This operation doesn't change the triangular structure of the matrix. In other words, if you start with an upper triangular matrix and perform this operation, you'll still have an upper triangular matrix, and the same goes for lower triangular matrices. So, if you apply this row operation to an upper triangular matrix, it remains an upper triangular matrix, and if you apply it to a lower triangular matrix, it remains a lower triangular matrix. There's no issue with doing this operation in the context of upper and lower triangular matrices; you're just modifying the values within the matrix without altering its triangular structure.
Best video
Every single video of tthis just shows with the right numbers. What needs to be done if you cant make a zero so easly? Where to multiply?
When dealing with upper and lower triangular matrices, the idea is to generate zeros in specific positions to form either an upper triangular matrix or a lower triangular matrix. If you're unable to easily create zeros directly, you might need to use row operations such as multiplication and addition to achieve this.
Here are some methods to transform a matrix into upper or lower triangular form when you can't easily create zeros:
Gaussian Elimination: Use row operations like multiplying rows by a scalar and adding multiples of one row to another to create zeros in desired positions. Continue this process until the matrix takes on the desired upper or lower triangular form.
Elementary Matrices: Perform row operations using elementary matrices. An elementary matrix represents an elementary row operation (such as swapping rows, multiplying a row by a scalar, or adding a multiple of one row to another). Multiply your original matrix by these elementary matrices to transform it into the desired form.
LU Decomposition: Factorize a matrix into a product of a lower triangular matrix (L) and an upper triangular matrix (U). If you have a matrix A, such that A = LU, where L is lower triangular and U is upper triangular, you can use this factorization to solve systems of linear equations or represent the matrix in upper/lower triangular form.
When performing row operations or using elementary matrices, the key is to maintain the equality of the matrix transformations. If you multiply the original matrix by a certain matrix on the left, ensure you perform the same operation on the right to maintain equality.
Remember, matrices can be transformed in various ways using elementary row operations, and the choice of which operation to use at a particular step depends on the specific elements and structure of the matrix you're working with.
Just found your video and it was a good explanation on the conversion. I tried doing it myself by pausing the video and what I got for the determinant is -30. I didn’t perform all the same operations that you did but I was still able to get the matrix into upper triangular form. My confusion is coming from the fact that we aren’t getting back the same results. I did the same operation as you for R3 and R4 in the same exact order. I then went on to get -3 in column2 to become 0 by the operation “R3= 2*R3 + 3*R2” and the 4 in column 3 to 0 by the operation “R4=5*R4 + 4*R3”. What I ended up with is multiplying 1*2*-5*3 = -30 as the determinant. Can you please help by explaining to me where I went wrong because I need help.
Let's walk through the steps to ensure we understand where the discrepancy might be coming from in calculating the determinant.
Given the steps you've mentioned, it looks like you're transforming a matrix into upper triangular form to find the determinant. The determinant of a matrix can be calculated by transforming it to upper triangular form and then multiplying the diagonal elements, taking into account any row swaps and scaling factors used during row operations.
Let's denote the matrix as \( A \). Here's the general process:
1. **Row operations to achieve upper triangular form:**
- Only row swaps affect the determinant by a factor of \(-1\) each time.
- Multiplying a row by a scalar \( k \) scales the determinant by \( k \).
- Adding or subtracting multiples of one row to another does not change the determinant.
2. **Calculation of the determinant:**
- After converting \( A \) to an upper triangular matrix \( U \), the determinant of \( A \) is the product of the diagonal elements of \( U \), adjusted by the factors from row swaps and row scalings.
Let's verify the operations you performed. Assume your original matrix \( A \) is:
\[ A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{pmatrix} \]
Here are the steps you mentioned:
1. **Operations on \( R3 \) and \( R4 \):**
- You applied \( R3 = 2*R3 + 3*R2 \). This operation does not change the determinant.
- You applied \( R4 = 5*R4 + 4*R3 \). This operation does not change the determinant.
2. **Achieving zeros in column 2 and column 3:**
- After these operations, you should have an upper triangular matrix. Let's denote it as \( U \).
3. **Calculating the determinant:**
- The determinant of the upper triangular matrix \( U \) is the product of its diagonal elements.
Let's assume you ended up with the following upper triangular matrix \( U \) after all operations (this is an illustration, as I don't have the exact matrix):
\[ U = \begin{pmatrix}
u_{11} & u_{12} & u_{13} & u_{14} \\
0 & u_{22} & u_{23} & u_{24} \\
0 & 0 & u_{33} & u_{34} \\
0 & 0 & 0 & u_{44}
\end{pmatrix} \]
The determinant of \( U \) is:
\[ \text{det}(U) = u_{11} \cdot u_{22} \cdot u_{33} \cdot u_{44} \]
### Check for Row Swaps and Scaling Factors:
- If there were any row swaps, each swap would multiply the determinant by \(-1\).
- If there were any row scalings by a factor \( k \), the determinant would be scaled by \( \frac{1}{k} \).
Given your determinant calculation of \( -30 \), you mentioned multiplying factors \( 1 \times 2 \times -5 \times 3 \):
- Ensure that these factors correspond to the diagonal elements of the final upper triangular matrix.
- Verify if any row swaps occurred (which would introduce a \(-1\) factor for each swap).
- Check if there were any row scalings, which would adjust the determinant accordingly.
Without the exact initial matrix and the intermediate steps, it's challenging to pinpoint the exact error. However, common mistakes might include:
- Missing a row swap or a row scaling.
- Incorrect application of row operations leading to incorrect upper triangular form.
- Arithmetic errors during row operations.
Thank you for the explanation. The video was really helpful. Cheers
this video was very helpful.❤
Saved me today Ty for such a helpful video
Nice 👍🏼👍🏼
Can the determinant will result a different answer when the way you eliminate numbers have different method to become zeros?
For example, your determinant A=3, and I had a different method to get the determinant, and I got A=72, is it still correct?
No, if you follow different methods to compute the determinant of the same matrix, the result must always be the same. The determinant is a unique scalar value associated with a square matrix, and its value does not depend on the method used, as long as all the steps follow valid mathematical rules.
In your example, if the determinant of matrix
𝐴
A is calculated to be 3, then any correct method of calculating it must also result in 3. If you use a different method and obtain 72, then an error likely occurred in your calculations. The steps for calculating the determinant (such as row reduction, cofactor expansion, or other techniques) must all lead to the same result.
If you get different results using different methods, it's important to review your calculations to ensure all steps were done correctly.
@@yesdevThank you. I lost it when I forgot to make all 1 diagonal. Isn't that the rule? Or it's a different rule from Pivoting?
There is no strict rule that the diagonal must be all 1s for determinant calculation. That’s more relevant for achieving reduced row echelon form.
The determinant is invariant under row addition/subtraction but changes sign when rows are swapped and scales if rows are multiplied by constants.
Pivoting and Gaussian elimination focus on numerical stability rather than affecting the determinant directly, except for the mentioned row swaps or scaling.
why wouldnt you multiply 3 to the determinant if you multiplied it in the row
When finding the determinant of a matrix using row operations, the determinant is multiplied by a scalar factor whenever a row operation is performed. However, it is important to remember that the determinant is a measure of the "scale factor" of the linear transformation represented by the matrix.
Multiplying a row of a matrix by a scalar factor changes the scale of the linear transformation represented by the matrix. If you multiply a row by 3, for example, you are effectively scaling the linear transformation by a factor of 3 along that particular row.
However, if you multiply the determinant by 3, you are effectively scaling the entire linear transformation by a factor of 3, which is not the same as just scaling one row of the matrix. This will result in an incorrect value for the determinant, since it is not measuring the scale factor of the original linear transformation.
Therefore, you should only multiply the determinant by a scalar factor that corresponds to the row operation being performed, not a scalar factor that applies to the entire matrix.
@@yesdev I don't think it answers his question, you multiplied a row by 3 in your calculation, why didn't you multiply it by the determinant of the resulting matrix? More explanations
please✨🤲
yeah I don’t get that part too
I updated my answer with more details i hope it helped
@@yesdev uhh i dont understand i wish you had a video that explained this part better
Pls make more videos 😍😍
yeah sure just subscribe amd notifications to get new notifications about last videos 😁
@@yesdev yeah I just did ❤❤❤
Great❤️
What is this method called?
Is this the same as gauss elimination method?
is Gaussian elimination, which is also known as row reduction or row echelon form. these matrices can be obtained through various methods such as LU decomposition, Cholesky decomposition, or simply by performing row operations on the original matrix. However, there isn't a specific method named after "upper and lower triangular matrices" themselves. They are usually produced as intermediate steps or results of other matrix operations
wait why didn;t u just do R3' = R3+(3/2) R2
In the context of performing row operations on matrices, it's important to consider the properties and structure of upper and lower triangular matrices.
An upper triangular matrix is a square matrix in which all entries below the main diagonal (from the top left to the bottom right) are zero. A lower triangular matrix is a square matrix in which all entries above the main diagonal are zero.
When you perform a row operation like R3' = R3 + (3/2)R2, you are adding a multiple of one row to another row. This operation doesn't change the triangular structure of the matrix. In other words, if you start with an upper triangular matrix and perform this operation, you'll still have an upper triangular matrix, and the same goes for lower triangular matrices.
So, if you apply this row operation to an upper triangular matrix, it remains an upper triangular matrix, and if you apply it to a lower triangular matrix, it remains a lower triangular matrix. There's no issue with doing this operation in the context of upper and lower triangular matrices; you're just modifying the values within the matrix without altering its triangular structure.