Determinants and Matrices

In linear algebra, determinants and matrices are used to solve linear equations by applying Cramer’s rule to a set of nonhomogeneous equations which are in linear form. Determinants are calculated for square matrices only. If the determinant of a matrix is zero, it is called as a singular determinant and if it is one, then it is known as unimodular. For the system of equations to have a unique solution, the determinant of the matrix must be nonsingular, that is its value must be nonzero.

Definition of a Determinant

A determinant can be defined in many ways for a square matrix.

The first and the most simplest way is to formulate the determinant by taking into account the top row elements and the corresponding minors. Take the first element of the top row and multiply it by its minor, then subtract the product of the second element and its minor. Continue to alternately add and subtract the product of each element of the top row with its respective minor until all the elements of the top row have been considered.

For example let us consider a 4×4 matrix A.

A = \(\begin{bmatrix} m & n & o & p\\ q & r & s & t\\ u & v & w & x\\ y & z & a & b \end{bmatrix}\)

Now its determinant |A| is defined as

|A| = \(\begin{vmatrix} m & n & o & p\\ q & r & s & t\\ u & v & w & x\\ y & z & a & b \end{vmatrix}\)

= \(m\begin{vmatrix} r & s & t\\ v & w & x\\ z & a & b \end{vmatrix}-n\begin{vmatrix} q & s & t\\ u & w & x\\ y & a & b \end{vmatrix}+o\begin{vmatrix} q & r & t\\ u & v & x\\ y & z & b \end{vmatrix}-p\begin{vmatrix} q & r & s\\ u & v & w\\ y & z & a \end{vmatrix}\)

The second way to define a determinant is to express in terms of the columns of the matrix by expressing an nXn matrix in terms of the column vectors.

Consider the column vectors of matrix A as A = [ a1, a2, a3, …an] where any element aj is a vector of size x.

Then the determinant of matrix A is defined such that

Det [ a1 + a2 …. baj+cv … ax ] = b det (A) + c det [ a1+ a2 + … v … ax ]

Det [ a1 + a2 …. aj aj+1… ax ] = – det [ a1+ a2 + … aj+1 aj … ax ]

Det (I) = 1

Where the scalars are denoted by b and c, a vector of size x is denoted by v, and the identity matrix of size x is denoted by I.

We can infer from these equations that the determinant is a linear function of the columns. Further, we observe that the sign of the determinant can be interchanged by interchanging the position of adjacent columns. The identity matrix of the respective unit scalar is mapped by the alternating multilinear function of the columns. This function is the determinant of the matrix.

Properties of a Determinant

If In is the identity matrix of the order nXn, then det(I) = 1

If the matrix MT is the transpose of matrix M, then det (MT) = det (M)

If matrix M-1 is the inverse of matrix M, then det (M-1) = \(\frac{1}{det (M)}\) = det (M)-1

If two square matrices M and N have the same size, then det (MN) = det (M) det (N)

If matrix M has a size aXa and C is a constant, then det (CM) = Ca det (M)

If X, Y, and Z are three positive semidefinite matrices of equal size, then the following holds true along with the corollary det (X+Y) \(\geq\) det(X) + det (Y) for X,Y, Z \(\geq\)0 det (X+Y+Z) + det C \(\geq\) det (X+Y) + det (Y+Z)

In a triangular matrix, the determinant is equal to the product of the diagonal elements.

The determinant of a matrix is zero if all the elements of the matrix are zero.

Laplace’s Formula and the Adjugate Matrix

With Laplace’s formula, the determinant of a matrix can be expressed in terms of the minors of the matrix.

If matrix Bxy is the minor of matrix A obtained by removing xth and yth column and has a size of

( j-1 X j-1), then the determinant of the matrix A is given by

det (A) = \(\sum_{y=1}^{j}(-1)^{x+y}a_{x,y}B_{x,y}\)

And \((-1)^{x+y}B_{x,y}\) is known as the cofactor.

The adjugate matrix is obtained by transposing the matrix containing the cofactors and is given by the equation,

(Adj (A))x,y = (-1)x+y Bx,y

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A triangle can have two: