Questions and Answers
If two matrices A and B have determinants 2 and 3, respectively, what is the determinant of the matrix product AB?
If a row of a matrix is added to another row, what happens to the determinant of the matrix?
What is the determinant of the inverse of a matrix with a determinant of 2?
If a matrix has a zero row, what is its determinant?
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What is the determinant of the transpose of a matrix with a determinant of 2?
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Study Notes
Properties of Determinants
Multiplicativity
- The determinant of the product of two matrices is the product of their determinants:
det(AB) = det(A) * det(B)
- This property can be extended to the product of multiple matrices
Linearity in Rows/Columns
- If two rows/columns of a matrix are interchanged, the determinant changes sign
- If a row/column of a matrix is multiplied by a scalar, the determinant is also multiplied by that scalar
- If a row/column of a matrix is added to another row/column, the determinant does not change
Homogeneity
- If all elements of a row/column of a matrix are multiplied by a scalar, the determinant is also multiplied by that scalar
Invariance Under Elementary Operations
- The determinant does not change under elementary row operations (swap, multiply, add)
- The determinant does not change under elementary column operations (swap, multiply, add)
Zero Determinant
- If a matrix has a zero row or column, its determinant is zero
- If a matrix has a linearly dependent row or column, its determinant is zero
Determinant of Inverse
- The determinant of the inverse of a matrix is the reciprocal of its determinant:
det(A^(-1)) = 1 / det(A)
Determinant of Transpose
- The determinant of the transpose of a matrix is the same as the determinant of the original matrix:
det(A^T) = det(A)
Properties of Determinants
Multiplicativity
- Determinant of a product of two matrices is equal to the product of their individual determinants
- This property can be extended to the product of multiple matrices
Linearity in Rows/Columns
- Interchanging two rows or columns of a matrix results in a change of sign of the determinant
- Multiplying a row or column by a scalar results in the determinant being multiplied by that scalar
- Adding a row or column to another row or column does not change the determinant
Homogeneity
- Multiplying all elements of a row or column by a scalar results in the determinant being multiplied by that scalar
Invariance Under Elementary Operations
- Elementary row operations (swap, multiply, add) do not change the determinant
- Elementary column operations (swap, multiply, add) do not change the determinant
Zero Determinant
- A matrix with a zero row or column has a determinant of zero
- A matrix with linearly dependent rows or columns has a determinant of zero
Determinant of Inverse and Transpose
- The determinant of the inverse of a matrix is the reciprocal of its determinant
- The determinant of the transpose of a matrix is equal to the determinant of the original matrix
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Description
This quiz covers the properties of determinants, including multiplicativity and linearity in rows and columns. Learn how to calculate determinants and understand their applications in linear algebra.