Determinants Flashcards

Vocabulary flashcards about determinants in linear algebra.

Determinant

A square arrangement of terms.

Element of a Determinant

Represents a value in the determinant; identified by its row and column position.

Order of a Determinant

The number of rows or columns in a determinant (m x n).

Expansion of a 2x2 Determinant

Calculated as (ad - cb) for a matrix [[a, b], [c, d]].

Expansion of a 3x3 Determinant

Involves expanding along a row or column using minors and cofactors.

Minor of an Element

The determinant of the matrix formed by deleting the row and column containing that element.

Cofactor of an Element

The minor of the element multiplied by (-1)^(i+j), where i and j are the row and column indices.

Value of Determinant Using Cofactors

Found by multiplying each element of a row or column by its corresponding cofactor and summing the results.

Transpose of a Determinant

Exchanging rows into respective columns; the determinant's value remains the same.

Interchanging Rows or Columns

Swapping any two rows or columns of a determinant changes the sign of its value.

Identical Rows or Columns

If any two rows or columns of a determinant are identical, its value is zero.

Multiplying a Row or Column by a Constant

If the elements of any row or column are multiplied by a constant (k), the determinant is multiplied by k.

Elementary Operation

Adding or subtracting a multiple of one row/column to another does not change the determinant's value.

Determinant as Sum of Determinants

If elements of a row/column are expressed as sums of two terms, the determinant can be expressed as the sum of two determinants.

Determinant with Elements in A.P.

If elements in rows or columns are in arithmetic progression, the determinant's value is zero.

System of Linear Equations - Cramer's Rule

A method using determinants to solve systems of linear equations.

Cramer's Rule - Unique Solution

If the determinant (Δ) is not equal to zero, the system has a unique solution (consistent system).

Homogeneous System

A system of equations where all constant terms are zero.

Homogeneous System - Trivial Solution

When Δ ≠ 0, the system has only the trivial solution (x=y=z=0).

Area of a Triangle Using Determinants

Given vertices (x1, y1), (x2, y2), (x3, y3), the area is calculated using a determinant formula.

Condition for Collinearity

Points are collinear if the area of the triangle formed by them is zero.

Differentiation of Determinants

Differentiating a determinant involves differentiating one row or column at a time.