Duplicate row determinant | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

Duplicate rows in a matrix result in a determinant of 0, indicating that the matrix is not invertible.

Transcript

Say I have some matrix a -- let's say a is n by n, so it looks something like this. You've seen this before, a 1 1, a 1 2, all the way to a 1 n. When you go down the rows you get a 2 1, that goes all the way to a 2 n. And let's say that there's some row here, let's say row i, it looks like a i 1, all the way to a i n. And then you have some other r... Read More

Key Insights

• 🤨 Matrices can be represented by row vectors, simplifying their notation.
• 🤨 Swapping two rows in a matrix creates a swap matrix, where the determinant is the negative of the original determinant.
• 🟰 If two rows in a matrix are equal, swapping them will result in the same matrix and a determinant equal to its negative value.

Questions & Answers

Q: What happens when two rows in a matrix are swapped?

Swapping two rows in a matrix creates a swap matrix, where the new matrix retains all the elements except for the swapped rows. The determinant of the swap matrix is the negative of the determinant of the original matrix.

Q: What if two rows in a matrix are the same?

If two rows in a matrix are identical, swapping them will not change the matrix. Therefore, the determinant of the matrix will be the same as its negative value.

Q: How does the determinant of a matrix with duplicate rows relate to its invertibility?

If a matrix has duplicate rows, it is not invertible because its reduced row echelon form cannot be transformed into the identity matrix. This is equivalent to the determinant of the matrix being equal to 0.

Q: Can duplicate columns in a matrix also result in a determinant of 0?

Yes, duplicate columns in a matrix will also lead to a determinant of 0, similar to duplicate rows. Additionally, if some rows in a matrix are linear combinations of other rows, the determinant will also be 0.

Summary & Key Takeaways

• Matrices can be represented by rows and columns, and by defining row vectors, the matrix can be simplified.

• Swapping two rows in a matrix creates a swap matrix, and the determinant of the swap matrix is the negative of the original determinant.

• If two rows in a matrix are equal, swapping them will result in the same matrix, and the determinant of the matrix will be equal to its negative value.