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Geometry and Algebra

The determinant: geometric meaning, Laplace expansion and Cramer's rule

Sarrus and Laplace, the determinant as area and volume, properties and Binet's theorem, computing with Gauss, invertibility, the adjugate and a proof of Cramer's rule.

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The determinant is a number attached to every square matrix that sums up a key fact: whether the matrix is invertible or not. It also has a precise geometric meaning, which makes all its properties natural: it measures how much the matrix stretches areas and volumes. In this lesson we define it, see how to compute it and look at its uses, up to Cramer’s rule.

Order 2 and 3

    \[ \det\begin{pmatrix} a & b\\ c & d \end{pmatrix} = ad - bc \]

For order 3 you can use the rule of Sarrus: add the products of the three diagonals going down to the right and subtract those of the three going down to the left.

    \[ \det\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix} = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} \]

Careful: Sarrus works only for 3×3 matrices. Higher orders need Laplace expansion or Gaussian elimination.

The geometric meaning

The columns of A are the images of the basis vectors, A\mathbf e_1 and A\mathbf e_2. The unit square built on \mathbf e_1, \mathbf e_2 is mapped to the parallelogram built on the columns of A.

The unit square mapped by a matrix to a parallelogram whose area equals the determinant
The matrix A maps the square of area 1 to the parallelogram built on its columns. The area of that parallelogram is |\det A|: here A has columns (2,\ 0.5) and (0.7,\ 1.8), and \det A = 3.6 - 0.35 = 3.25.

Proof in the 2×2 case

The parallelogram with sides (a, c) and (b, d) has an area equal to the magnitude of the cross product of the two vectors seen in space with a zero third component: (a, c, 0)\times(b, d, 0) = (0, 0, ad - bc). So the area is |ad - bc|. In three dimensions, likewise, |\det A| is the volume of the parallelepiped built on the columns: it is the triple product of the lesson on vectors.

The sign gives the orientation: positive if the transformation keeps the sense of rotation (from \mathbf e_1 to \mathbf e_2 anticlockwise), negative if it reverses it, as a mirror does. And if \det A = 0 the parallelogram is flattened onto a line: the transformation loses a dimension and cannot be inverted.

Laplace expansion

For an n\times n matrix, let A_{ij} be the submatrix obtained by deleting row i and column j. The cofactor is C_{ij} = (-1)^{i+j}\det A_{ij}. Then, expanding along any row i:

    \[ \det A = \sum_{j=1}^{n} a_{ij}\,C_{ij} \]

and the same holds along any column. Choose the row or column with the most zeros. The signs (-1)^{i+j} follow the chessboard pattern starting with + at the top left.

Example.

    \[ \det\begin{pmatrix} 2 & 0 & 1\\ 3 & 1 & 0\\ 1 & 4 & 5 \end{pmatrix} = 2\det\begin{pmatrix} 1 & 0\\ 4 & 5 \end{pmatrix} - 0 + 1\cdot\det\begin{pmatrix} 3 & 1\\ 1 & 4 \end{pmatrix} = 2\cdot 5 + 11 = 21 \]

Properties

  1. Swapping two rows changes the sign of the determinant.
  2. Multiplying a row by \lambda multiplies the determinant by \lambda. Hence \det(\lambda A) = \lambda^n\det A.
  3. Adding a multiple of one row to another does not change the determinant.
  4. If two rows are equal or proportional, or a row is zero, \det A = 0.
  5. \det A^T = \det A: everything true for rows is true for columns.
  6. The determinant of a triangular matrix is the product of its diagonal entries.
  7. Binet’s theorem: \det(AB) = \det A\cdot\det B.

Property 7 has an immediate geometric reading: applying B and then A scales areas first by \det B and then by \det A. It follows that \det(A^{-1}) = 1/\det A.

Computing with Gauss

Properties 1, 3 and 6 give the most efficient method: reduce the matrix to triangular form with elementary operations, multiply the diagonal and change sign once for each row swap. For a 10×10 matrix Laplace expansion takes millions of products, Gauss a few hundred.

Determinant and invertibility

Theorem. A square matrix A is invertible if and only if \det A \ne 0.

If A is invertible, by Binet \det A\cdot\det A^{-1} = \det I = 1, so \det A \ne 0. Conversely, if \det A \ne 0 the inverse can be written explicitly with cofactors:

    \[ A^{-1} = \frac{1}{\det A}\,\big(C_{ij}\big)^T \]

(the transpose of the cofactor matrix is called the adjugate). For 2×2 matrices we get back the formula from the lesson on matrices.

Cramer’s rule

If \det A \ne 0, the system A\mathbf x = \mathbf b has the single solution

    \[ x_i = \frac{\det A_i}{\det A} \]

where A_i is the matrix A with column i replaced by \mathbf b.

Proof. Let \mathbf x be the solution. In the matrix A_i column i is \mathbf b = x_1\mathbf a_1 + \dots + x_n\mathbf a_n, a combination of the columns of A. Since the determinant is linear in each column, \det A_i splits into n determinants: those with \mathbf a_j, j \ne i, in position i have two equal columns and are zero. What is left is \det A_i = x_i\det A.

Example. 2x + y = 5, x - 3y = -1: \det A = -6 - 1 = -7, x = \det\begin{pmatrix}5 & 1\\ -1 & -3\end{pmatrix}/(-7) = (-15 + 1)/(-7) = 2, y = \det\begin{pmatrix}2 & 5\\ 1 & -1\end{pmatrix}/(-7) = (-2 - 5)/(-7) = 1.

Cramer is handy for 2×2 and 3×3 systems by hand, and for getting symbolic formulas (like voltage dividers in circuits), but with many unknowns it is far slower than Gauss.

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