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Questions and Answers
What is the common notation for the determinant of a matrix A?
What is the common notation for the determinant of a matrix A?
When is the determinant of a matrix nonzero?
When is the determinant of a matrix nonzero?
How is the determinant of a 2 x 2 matrix calculated?
How is the determinant of a 2 x 2 matrix calculated?
Which formula expresses the determinant as a sum of n signed products of matrix entries?
Which formula expresses the determinant as a sum of n signed products of matrix entries?
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How can the determinant be computed using elementary row operations?
How can the determinant be computed using elementary row operations?
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Study Notes
Determinant Notation
- The determinant of a matrix A is commonly denoted by |A| or det(A).
Nonzero Determinant
- A matrix has a nonzero determinant if and only if it is invertible.
2x2 Matrix Determinant
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The determinant of a 2x2 matrix
A = [ a b ] [ c d ]is calculated as |A| = ad - bc.
Determinant Formula
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The determinant of an n x n matrix can be expressed as a sum of n signed products of matrix entries using the following formula:
|A| = Σ(σ∈Sn) sign(σ) a₁σ(₁) a₂σ(₂) ... anσ(n)where Sn is the set of all permutations of {1, 2, ..., n}, σ is a permutation, and sign(σ) is the sign of the permutation.
Computing Determinant Using Elementary Row Operations
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The determinant of a matrix can be computed using elementary row operations, which can modify the determinant in the following ways:
- Swapping two rows multiplies the determinant by -1.
- Multiplying a row by a scalar k multiplies the determinant by k.
- Adding a multiple of one row to another row does not change the determinant.
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Description
Test your knowledge about determinants, scalar values that represent a square matrix's properties and the linear map it represents. Explore the relationship between determinants and matrix invertibility.
