Linear Algebra · Chapter 04

Determinants

One number from a square matrix: how to compute it at every size, what it measures, how row operations change it, and how cofactors break it into smaller ones.

det (A)

a c b d = a d - b c

det (A B) = det (A) det (B)

det (A^{T}) = det (A)

C_{ij} = (- 1)^{i + j} M_{ij}

det (A) = j \sum a_{ij} C_{ij}

det (I) = 1

det (A) \neq = 0 ⟺ A^{- 1} exists

det (A - λ I)

Lessons

Build the foundation

01What Is a Determinant?One number from a square matrix: how to compute it at every size, what it measures, how row operations change it, and how cofactors break it into smaller ones.Read lesson →
02Determinants of 3×3 MatricesOne number from a square matrix: how to compute it at every size, what it measures, how row operations change it, and how cofactors break it into smaller ones.Read lesson →
03Area, volume and orientationA 2 by 2 determinant is an area. The columns of the matrix are two arrows, and the determinant measures the parallelogram they span, with a sign that says which way round they turn. For 3 by 3 matrices the same number is a volume.Read lesson →
04Determinants of any sizeEvery determinant is built the same way. Pick one entry from each row, never two from the same column, multiply them, and give the product a sign. Add all such signed products: two for a 2 by 2 matrix, six for 3 by 3, twenty-four for 4 by 4.Read lesson →
05Row operations and the determinantThe three row operations of elimination change a determinant in three simple ways. Swapping two rows flips its sign, scaling a row scales it, and adding a multiple of one row to another leaves it alone. So a determinant can be found by reducing the matrix to a triangle.

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06Properties of determinantsThree facts make the determinant useful. It does not change when a matrix is transposed, it turns a product of matrices into a product of numbers, and it is never zero for a matrix that can be undone. Each has a picture: area factors multiply when maps are composed.Read lesson →

07Minors and cofactorsA large determinant can be broken into smaller ones. Choose an entry, cross out its row and its column, and a smaller square matrix is left. Its determinant, with a sign that depends only on the position, is the cofactor of that entry.Read lesson →

08Laplace expansionA determinant can be expanded along any row or any column, not only the first: multiply each entry of that line by its cofactor and add. The skill is in choosing the line with the most zeros, since a zero entry takes its whole cofactor out of the work.Read lesson →

09Chapter practiceBring the chapter together: compute determinants of every size, read their geometry, use the row rules and the product rule, and expand along the best line. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Determinants

Build the foundation