Determinant when row is added | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

The determinants of matrices with identical rows except for one row that is the sum of the other two rows have a special relationship.

Transcript

Let's keep messing with our determinants to see if we can get more useful results. And they might not be obviously useful right now, but maybe we'll use them later when we are exploring other parts of linear algebra. So let's say I have some matrix, let's call it matrix X. Matrix X is equal to-- I'll just start with a 3 by 3 case because I think th... Read More

Key Insights

• 🤨 Matrices X, Y, and Z have a special relationship when one row in Z is the sum of the corresponding rows in X and Y.
• 💤 The determinants of X, Y, and Z show that the determinant of Z is the sum of the determinants of X and Y.
• 🤨 This relationship only holds when the matrices are identical except for one specific row.
• 💤 The relationship between the determinants of X, Y, and Z can be generalized to matrices of any dimension.

Questions & Answers

Q: What is the relationship between the determinants of matrices X, Y, and Z?

The determinant of Z is equal to the determinant of X plus the determinant of Y, given that one row in matrix Z is the sum of the corresponding rows in X and Y.

Q: In what case does the relationship between the determinants of X, Y, and Z hold?

The relationship holds when the matrices are identical except for one specific row, which is the sum of the corresponding rows in X and Y.

Q: Can the relationship between the determinants of X, Y, and Z be generalized to matrices with different dimensions?

Yes, the relationship holds for matrices of any dimension, as long as the matrices are identical except for one specific row.

Q: Is it always true that the determinant of Z is equal to the determinant of X plus the determinant of Y if Z is equal to X plus Y?

No, the relationship only holds when matrices X, Y, and Z are identical everywhere except for one specific row. Determinant operations are not linear on matrix addition in general.

Summary & Key Takeaways

• Matrices X, Y, and Z are defined, where X and Y are identical except for one row, and Z has the same first row as X and Y but the second row is the sum of the second rows of X and Y.

• The determinants of X, Y, and Z are calculated, showing that the determinant of Z is equal to the determinant of X plus the determinant of Y.

• This relationship only holds when the matrices are identical everywhere except for one specific row.