Determinants, their properties, and computation methods from a college linear algebra course.
30 cards · basic cards · AI-written, checked twice. Edit anything.
- What is the determinant of a square matrix?
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A scalar value computed from a square matrix's entries that encodes information about the matrix's invertibility, scaling effect, and orientation.- What is the notation for the determinant of matrix A?
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det(A) or |A|- State the formula for the determinant of a 2 by 2 matrix [[a, b], [c, d]].
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ad - bc- Describe the general formula for a 3 by 3 determinant using the rule of Sarrus or expansion.
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Sum of products along three diagonals minus sum of products along three anti-diagonals (Sarrus), or expansion along a row or column of minors and cofactors.- What does the absolute value of a determinant represent geometrically?
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The volume (or area in 2D) of the parallelepiped (or parallelogram) formed by the matrix's row or column vectors.- What is the determinant of the n by n identity matrix?
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1- A square matrix is invertible if and only if its determinant is ___.
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nonzero- If two rows of a matrix are swapped, how does the determinant change?
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The determinant is multiplied by -1.- If one row of a matrix is multiplied by a scalar c, how does the determinant change?
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The determinant is multiplied by c.- If a multiple of one row is added to another row, how does the determinant change?
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The determinant does not change.- What is a minor of a matrix?
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The determinant of the submatrix obtained by deleting one row and one column.- What is a cofactor of a matrix?
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The minor of a position multiplied by (-1)^(i+j), where i and j are the row and column indices.- State the cofactor expansion formula for the determinant along row i.
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det(A) = sum over j of a(i,j) * C(i,j), where C(i,j) is the cofactor of position (i,j).- What is the relationship between det(A) and det(A^T)?
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det(A^T) = det(A); the determinant of a matrix equals the determinant of its transpose.- State the product rule for determinants.
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det(AB) = det(A) * det(B)
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