← Linear Algebra

Unit 6: Determinants

One number that measures how much a matrix scales area or volume, flips orientation, or collapses space entirely.

8 topics · 29 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

What a Determinant Measures

  1. How it worksHow does the determinant track the area of the parallelogram formed by a matrix's columns?
  2. Worked exampleFind the area of the parallelogram formed by the vectors (3, 1) and (1, 2) using the determinant of the matrix they form.
  3. Common mistakeWhy doesn't a determinant of zero mean the matrix itself is empty or has no entries?
  4. Going furtherWhat happens to the shape of that parallelogram once the determinant turns out negative?
Your turn1 of 2
Find the area of the parallelogram formed by the vectors (4, 3) and (8, -4).

The 2x2 and 3x3 Determinant Formulas

  1. Worked exampleFind the determinant of the matrix [[3, 1], [2, 4]].
  2. Worked exampleFind the determinant of the 3 by 3 matrix [[1, 2, 3], [0, 1, 4], [5, 6, 0]] using the full expansion.
  3. Common mistakeWhy doesn't the 3 by 3 determinant formula just repeat the 2 by 2 pattern of ad minus bc three separate times?
  4. Going furtherWhat happens to the 2 by 2 determinant formula the moment two rows of the matrix become identical?
Your turn1 of 2
Find the determinant of the matrix [[4, 3], [8, -4]].

Properties of Determinants

  1. How it worksHow does each elementary row operation change a determinant in a completely predictable way?
  2. Worked exampleFind the determinant of [[2, 4], [1, 3]] after doubling the first row, and compare it with the original determinant.
  3. Common mistakeWhy doesn't the determinant of A + B usually equal the determinant of A plus the determinant of B?
  4. Going furtherWhat happens to a determinant once one row is a multiple of another row?
Your turn1 of 2
The matrix [[4, 3], [8, -4]] has determinant ad - bc. Find the determinant after doubling the first row.

Cofactor Expansion

  1. How it worksHow does expanding along a row multiply each entry by the determinant left after deleting its row and column?
  2. Worked exampleFind the determinant of the matrix [[1, 0, 2], [3, 1, 1], [0, 2, 1]] by cofactor expansion along the first row.
  3. Going furtherWhat happens to cofactor expansion once a whole row of the matrix is zero?
Your turn1 of 2
Find the determinant of the matrix [[2, 2, 5], [-3, -5, -4], [2, -6, -4]] by cofactor expansion along the second row.

The Determinant of a Product and an Inverse

  1. How it worksHow does det(AB) equal det(A) times det(B) fall directly out of what a determinant measures?
  2. Worked exampleFind det(A inverse) given that det(A) equals 5.
  3. Common mistakeWhy isn't the determinant of A + B related to det(A) and det(B) the same simple way multiplication is?
Your turn1 of 2
Given that det(A) equals 9, find det(A inverse).

The Determinant as a Test for Invertibility

  1. How it worksHow does a determinant of zero mean the matrix squashes space down and destroys information that can never come back?
  2. Worked exampleFind whether the matrix [[2, 4], [1, 2]] is invertible by checking its determinant.
  3. Common mistakeWhy doesn't a zero determinant automatically mean a system has no solution, when it might instead have infinitely many?
  4. Going furtherWhat happens to this determinant test for invertibility once the matrix isn't square at all?
Your turn1 of 2
Find the determinant of the matrix [[4, 3], [8, -4]]. The matrix is invertible exactly when this value is nonzero.

Cramer's Rule

  1. How it worksHow does replacing one column of A with b and comparing determinants isolate a single variable?
  2. Worked exampleSolve for x in the system 2x + y = 5, x - y = 1 using Cramer's rule.
  3. Going furtherWhat happens to Cramer's rule once one of the determinants in its formula turns out to be zero?
  4. Going furtherWhat happens to Cramer's rule when the determinant of the coefficient matrix itself is zero?
Your turn1 of 2
Solve for x in the system 7x + 2y = -9, 3x + 5y = -6 using Cramer's rule.

Determinants and Volume

  1. How it worksHow do three vectors in 3-space define a parallelepiped whose volume equals the determinant of the matrix they form?
  2. Worked exampleFind the volume of the parallelepiped spanned by (1, 0, 0), (0, 2, 0), and (1, 1, 3).
  3. Ties togetherHow does a zero determinant meaning zero volume connect back to the columns being linearly dependent?
Your turn1 of 2
Find the volume of the parallelepiped spanned by (4, 0, 0), (0, 3, 0), and (8, -4, -8).