Skip to content

Mathematics · Ch 4 — Determinants

Determinant

4.2

Determinant

4.2 Determinant

The determinant assigns a single number (real or complex) to every square matrix — formally a function f:M→Kf: M \to K from the set MM of square matrices to the set of numbers KK, written f(A)=∣A∣f(A) = |A|. It encodes key properties, such as whether the matrix is invertible. The determinant of AA is denoted ∣A∣|A|, det⁡(A)\det(A), or Δ\Delta (the Greek letter Delta).

For a 2×22 \times 2 matrix A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}:

∣A∣=∣abcd∣=det⁡(A)|A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = \det(A)

Note

∣A∣|A| for a determinant is not the modulus (absolute value); for a matrix, ∣A∣|A| always means the determinant.

Watch out

Only square matrices have determinants — a non-square matrix (e.g. a 2×32 \times 3 matrix) has none.

Determinant of a 2×22 \times 2 Matrix

For A=[a11a12a21a22]A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix},

det⁡(A)=∣A∣=a11a22−a21a12\det(A) = |A| = a_{11}a_{22} - a_{21}a_{12}

the product of the main-diagonal entries minus the product of the other diagonal.

Tip

For [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix}: the arrow top-left to bottom-right gives a×da \times d, and bottom-left to top-right gives c×bc \times b; subtract to get ad−cbad - cb.

Determinant of a 3×33 \times 3 Matrix

For A=[a11a12a13a21a22a23a31a32a33]A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}, expand along the first row:

∣A∣=a11∣a22a23a32a33∣−a12∣a21a23a31a33∣+a13∣a21a22a31a32∣|A| = a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}

Each term is a first-row element, times the sign (+,−,++, -, + across the columns), times the minor — the 2×22 \times 2 determinant left after deleting that element's row and column.

Important

The sign for the element in row ii, column jj is (−1)i+j(-1)^{i+j}; along the first row this gives +,−,++, -, +.

Properties of Determinants

Each property is stated and proved for a 2×22 \times 2 matrix (the book's approach); the reasoning extends to square matrices of any order.

›Proof

Property 1: det⁡(A)=det⁡(AT)\det(A) = \det(A^T).

For A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, AT=[acbd]A^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix}: det⁡(A)=ad−bc=ad−cb=det⁡(AT)\det(A) = ad - bc = ad - cb = \det(A^T). Holds for all square matrices.

›Proof

Property 2: Interchanging two rows (or columns) changes the sign of the determinant.

Interchanging the rows of A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} gives B=[cdab]B = \begin{bmatrix} c & d \\ a & b \end{bmatrix}: det⁡(B)=cb−da=−(ad−bc)=−det⁡(A)\det(B) = cb - da = -(ad - bc) = -\det(A). The same holds for columns.

›Proof

Property 3: Two identical rows (or columns) make the determinant zero.

For A=[abab]A = \begin{bmatrix} a & b \\ a & b \end{bmatrix}, det⁡(A)=ab−ab=0\det(A) = ab - ab = 0. Likewise for identical columns.

›Proof

Property 4: Multiplying a row (or column) by kk multiplies the determinant by kk.

Scaling the first row of A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} gives B=[kakbcd]B = \begin{bmatrix} ka & kb \\ c & d \end{bmatrix}: det⁡(B)=(ka)d−c(kb)=k(ad−bc)=kdet⁡(A)\det(B) = (ka)d - c(kb) = k(ad - bc) = k\det(A).

›Proof

Property 5: A row (or column) written as a sum splits the determinant into two.

For A=[a1+a2bcd]A = \begin{bmatrix} a_1 + a_2 & b \\ c & d \end{bmatrix}, det⁡(A)=(a1+a2)d−cb=(a1d−cb)+(a2d−cb)=∣a1bcd∣+∣a2bcd∣\det(A) = (a_1 + a_2)d - cb = (a_1 d - cb) + (a_2 d - cb) = \begin{vmatrix} a_1 & b \\ c & d \end{vmatrix} + \begin{vmatrix} a_2 & b \\ c & d \end{vmatrix}.

›Proof

Property 6: Adding a multiple of one row (or column) to another leaves the determinant unchanged. …

Start your 14-day free trial to unlock the full solution →
← Previous · 4.1IntroductionNext · 4.2.1 →Determinant of a Matrix of Order One