Podcast
Questions and Answers
What is the determinant of the product of two square matrices A and B?
What is the determinant of the product of two square matrices A and B?
If B is obtained by adding a multiple of one row of A to another row, what is the relation between det(A) and det(B)?
If B is obtained by adding a multiple of one row of A to another row, what is the relation between det(A) and det(B)?
What is the determinant of the scalar multiple of a matrix?
What is the determinant of the scalar multiple of a matrix?
What happens to the determinant of a matrix under elementary row operations?
What happens to the determinant of a matrix under elementary row operations?
Signup and view all the answers
What is the determinant of a matrix equivalent to A?
What is the determinant of a matrix equivalent to A?
Signup and view all the answers
What is the general formula for the determinant of the sum of two matrices?
What is the general formula for the determinant of the sum of two matrices?
Signup and view all the answers
Study Notes
Determinant Properties
Multiplicative Property
-
det(AB) = det(A)det(B)for square matrices A and B - This property allows us to break down the calculation of the determinant of a matrix product into the product of the determinants of the individual matrices.
Additive Property in Rows/Columns
-
det(A + B) ≠ det(A) + det(B)in general - However, if B is obtained by adding a multiple of one row/column of A to another row/column, then
det(A + B) = det(A) - This property is useful for finding the determinant of a matrix by using row/column operations.
Scalar Multiplication Property
-
det(kA) = k^n det(A)for a square matrix A and scalar k, where n is the size of the matrix - This property shows that the determinant of a matrix is scaled by a factor of k^n when the matrix is multiplied by a scalar k.
Invariance under Elementary Row/Column Operations
- The determinant remains unchanged under elementary row/column operations:
- Swap two rows/columns:
det(A) = det(A') - Multiply a row/column by a non-zero scalar:
det(A) = det(A') - Add a multiple of one row/column to another row/column:
det(A) = det(A')
- Swap two rows/columns:
Invariance under Matrix Equivalence
- The determinant remains unchanged under matrix equivalence:
-
det(PAQ) = det(A)for invertible matrices P and Q - This property shows that the determinant is a property of the matrix itself, not just its representation.
-
Multilinearity
- The determinant is a multilinear function, meaning it is linear in each row/column separately:
-
det(A + B, C, ..., Z) = det(A, C, ..., Z) + det(B, C, ..., Z) -
det(kA, C, ..., Z) = k det(A, C, ..., Z)
-
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Learn about the properties of determinants, including multiplicative, additive, and scalar multiplication properties. Understand how determinants remain invariant under elementary row/column operations and matrix equivalence.
