Determinants: every key term you need (+ practice quiz)
25 flashcard terms for Linear Algebra Topic 6, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A scalar assigned to each square matrix, vanishing exactly when the matrix is singular and measuring the signed factor by which the matrix scales volume.
Cofactor expansion
A recursive formula computing a determinant along any chosen row or column as an alternating sum of entries times the determinants of smaller submatrices.
Minor of an entry
The determinant of the matrix left after deleting the row and column containing that entry; it is the building block of cofactor expansion.
Cofactor of an entry
The minor of the entry multiplied by a sign that alternates in a checkerboard pattern according to the parity of the sum of the row and column indices.
Expansion along any line
A theorem stating that cofactor expansion gives the same value regardless of which row or column is chosen, which makes rows with many zeros the efficient choice.
Determinant of a triangular matrix
The product of the diagonal entries. This makes row reduction to triangular form a practical way to evaluate large determinants.
Effect of row swaps
Interchanging two rows multiplies the determinant by negative one, so a matrix with two identical rows must have determinant zero.
Effect of row scaling
Multiplying a single row by a scalar multiplies the determinant by that same scalar, so scaling an entire n by n matrix multiplies it by the scalar to the nth power.
Effect of adding a multiple of a row
Adding a multiple of one row to another leaves the determinant unchanged, which is why elimination can be carried out without correcting the value.
Multiplicative property
The determinant of a product equals the product of the determinants. It follows that a matrix and its inverse have reciprocal determinants.
Determinant and the transpose
A matrix and its transpose have the same determinant, which is why every statement about rows has a matching statement about columns.
Singularity criterion
A square matrix is invertible exactly when its determinant is nonzero. This is one of the equivalent conditions of the Invertible Matrix Theorem.
Determinant as signed volume
The absolute value of the determinant is the volume of the parallelepiped spanned by the columns, and the sign records whether orientation is preserved.
Area scaling in the plane
For a linear map of the plane, the area of the image of any region equals the area of the region times the absolute value of the determinant.
Adjugate matrix
The transpose of the matrix of cofactors. Multiplying a matrix by its adjugate produces the determinant times the identity matrix.
Inverse by adjugate
The inverse of an invertible matrix equals its adjugate divided by its determinant, a formula of great theoretical value but poor computational efficiency.
Cramer rule
For an invertible square system, each unknown equals a ratio of determinants, the numerator coming from replacing one column of the coefficient matrix by the constants.
Alternating multilinear characterization
The determinant is the unique function of the columns that is linear in each column separately, changes sign under a swap, and equals one on the identity matrix.
Permutation formula
The determinant equals a sum over all orderings of the columns, each term a signed product of entries drawn one from each row and each column.
Sign of a permutation
Plus one for an even number of transpositions and minus one for an odd number; this sign supplies the alternating pattern in the permutation formula.
Determinant of a block triangular matrix
When the blocks below or above the diagonal vanish, the determinant is the product of the determinants of the square diagonal blocks.
Vanishing determinant from dependence
If the columns of a square matrix are dependent, some column is a combination of the others and the determinant is forced to zero by multilinearity.
Determinant of a scalar multiple
Multiplying an n by n matrix by a scalar multiplies its determinant by that scalar raised to the power n, since every row is scaled.
Determinant under similarity
Conjugate matrices have equal determinants because the factors from the conjugating matrix and its inverse cancel by the multiplicative property.
Wronskian style test
Arranging vectors as columns of a square matrix and computing the determinant gives a quick independence test: the vectors are independent exactly when it is nonzero.