Determinants

4.1 Determinants of Order 2

TipDeterminant of Order 2

If \[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \]

is a \(2 \times 2\) matrix with entries from a field \(F\), then we define the determinant of \(A\), denoted \(\det(A)\) or \(|A|\), to be the scalar

\[ ad - bc. \]

The function \(\det : M_{2\times2}(F) \to F\) is a linear function of each row of a \(2\times2\) matrix when the other row is held fixed. That is, if \(u, v,\) and \(w\) are in \(F^2\) and \(k\) is a scalar, then \[ \det \begin{pmatrix} u + kv \\ w \end{pmatrix} = \det \begin{pmatrix} u \\ w \end{pmatrix} + k \det \begin{pmatrix} v \\ w \end{pmatrix}. \]

and

\[ \det \begin{pmatrix} w \\ u + kv \end{pmatrix} = \det \begin{pmatrix} w \\ u \end{pmatrix} + k \det \begin{pmatrix} w \\ v \end{pmatrix}. \]

4.2 Determinants of Order n

TipDeterminant of Order n

For this definition, it is convenient to introduce the following notation: Given \(A \in M_{n\times n}(F)\), for \(n \ge 2\), denote the \((n-1)\times(n-1)\) matrix obtained from \(A\) by deleting row \(i\) and column \(j\) by \(\tilde{A}_{ij}\).

Let \(A \in M_{n\times n}(F)\). If \(n = 1\), so that \(A = (A_{11})\), we define \[ \det(A) = A_{11}. \]

For \(n \ge 2\), we define \(\det(A)\) recursively as \[ \det(A) = \sum_{j=1}^{n} (-1)^{1+j} A_{1j} \cdot \det(\tilde{A}_{1j}). \]

The scalar \(\det(A)\) is called the determinant of \(A\) and is also denoted by \(|A|\). The scalar \[ (-1)^{i+j} \det(\tilde{A}_{ij}) \] is called the cofactor of the entry of \(A\) in row \(i\), column \(j\).

• Letting \[ c_{ij} = (-1)^{i+j} \det(\tilde{A}_{ij}) \]

  denote the cofactor of the row \(i\), column \(j\) entry of \(A\), we can express the formula for the determinant of \(A\) as \[ \det(A) = A_{11} c_{11} + A_{12} c_{12} + \cdots + A_{1n} c_{1n}. \]

Theorem 4.1 The determinant of an \(n \times n\) matrix is a linear function of each row when the remaining rows are held fixed. That is, for \(1 \le r \le n\), we have \[ \det \begin{pmatrix} a_1 \\ \vdots \\ a_{r-1} \\ u + kv \\ a_{r+1} \\ \vdots \\ a_n \end{pmatrix} = \det \begin{pmatrix} a_1 \\ \vdots \\ a_{r-1} \\ u \\ a_{r+1} \\ \vdots \\ a_n \end{pmatrix} + k \det \begin{pmatrix} a_1 \\ \vdots \\ a_{r-1} \\ v \\ a_{r+1} \\ \vdots \\ a_n \end{pmatrix}. \]

whenever \(k\) is a scalar and \(u\), \(v\), and each \(a_i\) are row vectors in \(F^n\).

4.3 Properties of Determinants

Theorem 4.2 If \(A \in M_{n \times n}(F)\) has a row consisting entirely of zeros, then \[ \det(A) = 0. \]

Theorem 4.3 The determinant of a square matrix can be evaluated by cofactor expansion along any row. That is, if \(A \in M_{n \times n}(F)\), then for any integer \(i\) (\(1 \le i \le n\)), \[ \det(A) = \sum_{j=1}^{n} (-1)^{i+j} A_{ij} \cdot \det(\tilde{A}_{ij}). \]

Theorem 4.4 If \(A \in M_{n \times n}(F)\) and \(B\) is a matrix obtained from \(A\) by interchanging any two rows of \(A\), then \[ \det(B) = -\det(A). \]

Theorem 4.5 Let \(A \in M_{n \times n}(F)\), and let \(B\) be a matrix obtained by adding a multiple of one row of \(A\) to another row of \(A\). Then \[ \det(B) = \det(A). \]

Theorem 4.6 If \(A \in M_{n \times n}(F)\) has rank less than \(n\), then \[ \det(A) = 0. \]

Theorem 4.7 For any \(A, B \in M_{n \times n}(F)\), we have \[ \det(AB) = \det(A)\,\det(B). \]

Theorem 4.8 For any \(A \in M_{n \times n}(F)\), we have \[ \det(A^{t}) = \det(A). \]

Theorem 4.9 (Cramer’s Rule) Let \(Ax = b\) be the matrix form of a system of \(n\) linear equations in \(n\) unknowns, where \(x = (x_1, x_2, \ldots, x_n)^t\). If \(\det(A) \neq 0\), then the system has a unique solution, and for each \(k\) (\(k = 1,2,\ldots,n\)), \[ x_k = \frac{\det(M_k)}{\det(A)}, \]

where \(M_k\) is the \(n \times n\) matrix obtained from \(A\) by replacing column \(k\) of \(A\) with \(b\).

Theorem 4.10 For invertible \(A\), its inverse \(A^{-1}\) may be described as \[ (A^{-1})_{ij} = \frac{(-1)^{\,i+j} M_{ji}}{\det(A)} \]

where \(M_{ji}\) is the determinant of \(A\) deleting the \(j\)-th row and \(i\)-th column. Pay attention to the subscript inverse.