# MATLAB TUTORIAL, part 2.1: Determinant

Determinants

General definition of determinant (the term determinant was first introduced by the German mathematician Carl Friedrich Gauss in 1801) is difficult and counterintuitive. The determinant of a square $$n \times n$$ matrix A is the value that is calculated as the sum of n! terms, half of them are taken with sign plus, and another half has oposite sign. The concept of a determinant first appears, nearly two millennia before its supposed invention by the Japanese mathematician Seki Kowa (1642--1708) in 1683 or his German contemporary Gottfried Leibniz (1646--1716). Traditionally, the determinant of a square matrix is denoted by det(A), det A, or |A|.

We define it recursively using cofactor expansion. For a $$1 \times 1$$ matrix that consists of one element, $${\bf A} = [a] ,$$ its determinant is $$\det {\bf A} = \left\vert {\bf A} \right\vert = a .$$ For a $$2\times 2$$ matrix $${\bf A} = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right] ,$$ its determinant is $$\det {\bf A} = a_{11} a_{22} - a_{12} a_{21} .$$ Similarly, for a $$3 \times 3$$ matrix, we have a specific formula:

$\det \begin{bmatrix} a&b&c \\ d&e&f \\ g&h&i \end{bmatrix} = a\,\left\vert \begin{array}{cc} e&f \\ h&i \end{array} \right\vert - b\,\left\vert \begin{array}{cc} d&f \\ g&i \end{array} \right\vert + c \,\left\vert \begin{array}{cc} d&e \\ g&h \end{array} \right\vert = aei + bfg + cdh - ceg -bdi -afh .$

If A is a square matrix, then the minor of entry $$a_{ij}$$ is denoted by $${\bf M}_{ij}$$ and is defined to be the determinant of the submatrix that remains after the i-th row and j-th column are deleted from A. The number $$(-1)^{i+j} {\bf M}_{ij}$$ is denoted by $${\bf C}_{ij}$$ and is called the cofactor of entry $$a_{ij} .$$

matlab uses two standard commands to evaluate the determinant of a square matrix

			det(A)
Writing inv(A) or A^(-1) calculate the inverse:
			inv(A)
			A^(-1)

We list the main properties of determinants:

1. $$\det ({\bf I} ) = 1 ,$$ where I is the identity matrix (all entries are zeroes except diagonal terms, which all are ones).
2. $$\det \left( {\bf A}^{\mathrm T} \right) = \det \left( {\bf A} \right) .$$
3. $$\det \left( {\bf A}^{-1} \right) = 1/\det \left( {\bf A} \right) = \left( \det {\bf A} \right)^{-1} .$$
4. $$\det \left( {\bf A}\, \det {\bf B} \right) = \det {\bf A} \, \det {\bf B} .$$
5. $$\det \left( c\,{\bf A} \right) = c^n \,\det \left( {\bf A} \right)$$ for $$n\times n$$ matrix A and a scalar c.
6. If $${\bf A} = [a_{i,j}]$$ is a triangular matrix, i.e. $$a_{i,j} = 0$$ whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries:

$\det \left( {\bf A} \right) = a_{1,1} a_{2,2} \cdots a_{n,n} = \prod_{i=1}^n a_{i,i} .$

An $$n \times n$$ square matrix A is called invertible if there exists an $$n \times n$$ matrix B such that

${\bf A}\, {\bf B} = {\bf B}\,{\bf A} = {\bf I} ,$
where I is the identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A, denoted by $${\bf A}^{-1} .$$ A matrix that is its own inverse, i.e. $${\bf A} = {\bf A}^{-1}$$ and $${\bf A}^{2} = {\bf I},$$ is called an involution.

Its second power $${\bf B}\,{\bf B} = {\bf B}^2 = -{\bf I} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$$ is the square root of the negative identity matrix, $$-{\bf I} = -\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} .$$ While the fourth power of the matrix B is the identity matrix: $${\bf B}^4 = {\bf I} .$$

A matrix A is called singular if and only if its determinant is zero. Otherwise, the matrix is nonsingular or invertible (because an inverse matrix exists for such matrix). The Cayley–Hamilton method for $$2 \times 2$$ matrix gives
${\bf A}^{-1} = \frac{1}{\det {\bf A}} \left[ \left( \mbox{tr} {\bf A} \right) {\bf I} - {\bf A} \right] .$

We list some basic properties of inverse operation:

1. $$\left( {\bf A}^{-1} \right)^{-1} = {\bf A} .$$
2. $$\left( c\,{\bf A} \right)^{-1} = c^{-1} \,{\bf A}^{-1}$$ for nonzero scalar c.
3. $$\left( {\bf A}^{\mathrm T} \right)^{-1} = \left( {\bf A}^{-1} \right)^{\mathrm T} .$$
4. $$\left( {\bf A}\, {\bf B} \right)^{-1} = {\bf B}^{-1} {\bf A}^{-1} .$$

A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if $${\bf A}^{\mathrm T} = {\bf A}^{-1} .$$ A matrix that is its own inverse, i.e., $${\bf A} = {\bf A}^{-1}$$ is called an involution.


Complete