Determinants and Matrices Definition, Types, Properties

Determinants are scalar numbers formed by multiplying the elements of a square matrix by their cofactors according to a predefined method. They assist in determining the adjoint, or inverse, of a matrix. Additionally, we must apply this concept to solve linear equations using the matrix inversion method. The cross-product of two vectors is easily recalled using determinants.

Definition of Matrices

Matrices are rectangular arrays of numbers in an ordered fashion that are used to express linear equations. A matrix is made up of rows and columns of data. Additionally, we can execute mathematical operations on matrices, such as addition, subtraction, and multiplication. If the matrix has m rows and n columns, it is represented as a m x n matrix.

Definition of Determinant

For a square matrix, a determinant can be defined in a variety of ways.

The first and most straightforward method is to construct the determinant by considering the top row items and their corresponding minors. Multiply the first element in the top row by its minor, and then subtract the product of the second element and its minor from the result of the multiplication. Continue adding and subtracting the product of each element in the top row with its corresponding minor element in this manner until all components in the top row have been evaluated.

Properties of Determinants

A determinant is a specific number in linear algebra that can be computed from a square matrix. Denoted as det(P), |P|, or det P, the determinant of a matrix, say P, is det(P). Determinants possess several advantageous qualities that enable us to get the same outcomes using alternative and simpler configurations of entries (elements). There are ten primary qualities of determinants. They are reflection, all zero, proportionality or repetition, switching, scalar multiple, sum, invariance, factor, triangle, and cofactor matrix. Each determinant property has been discussed in length below, along with solved cases.

Significant Determinant Properties

1. Reflection Property

If the rows of the determinant are converted to columns and the columns to rows, the determinant remains unchanged. This is referred to as the reflection property.

2. All-zero Property

The determinant is 0 if all the elements of a row (or column) are zero.

3. Proportionality (Repetition) Property

If all of the items in a row (or column) are proportionate (identical) to those in another row (or column), the determinant is zero.

4. Switching Property

The sign of the determinant is changed when any two rows (or columns) of the determinant are interchanged.

5. Scalar Multiple Property

When all the elements of a determinant’s row (or column) are multiplied by a non-zero constant, the determinant itself is multiplied by the same constant.

Apart from these features, determinants possess additional properties, including the following:

• If In is the order n x n identity matrix, then det(I) = 1.
• If MT is the transpose of M, then det (MT) = det (M)
• If M-1 is the reciprocal of M, then det (M-1) = 1/det(M) = det (M)-1.
• If the dimensions of two square matrices M and N are equal, then det (MN) = det (M) (N)
• If matrix M has the dimension a x a and constant C, then det (CM) = Ca det (M)
• If X, Y, and Z are three positive semidefinite matrices of equal size, the following holds, along with the corollary det (X+Y) ≥ det(X) + det (Y) for X, Y, Z ≥ 0 det (X+Y+Z) + det C ≥ det (X+Y) + det (Y+Z)
• The determinant of a triangular matrix is equal to the product of the diagonal elements.
• A matrix’s determinant is 0 if all of its elements are zero.
• The Formula of Laplace and the Adjugate Matrix
• Property of Sum
• Property of the Factor
• Triangle Property
• Property of Invariance
• The determinant of the matrix Cofactor

Determinant of a Matrix

The determinant of the simplest square matrix of order 11, which contains only one integer, is the number itself. Let us study how to calculate the determinants for matrices of second, third, and fourth order.

The determinants are critical in solving systems of linear equations and determining the inverse of a matrix. Now, let’s look at how to determine the determinant of a 22 matrix and a 33 matrix. If A is a matrix, the determinant of A is typically det (A) or |A|.

Rules For Operations on Determinants

To execute row and column operations on determinants, the following rules are useful:

• If the rows and columns are swapped, the determinant value remains unchanged.

• If any two rows or (two columns) are swapped, the sign of the determinant changes.

• Any two rows or columns of a matrix have the same value if and only if they are equal, and the determinant has the value zero.

• Whenever the value of each element in a row or column is multiplied by a constant, the value of the determinant is also multiplied by the same constant.

• If a row or column’s elements are expressed as a sum of elements, the determinant can be expressed as a sum of determinants.

• When the elements of a row or column are added or subtracted from the elements of another row or column, the determinant value remains intact.

Determinants for the 2 x 2 matrix:

Calculate the determinant of any 2×2 square matrix or square matrix of order 2 x 2 using the determinant formula:

C =  |a b c d|

Its 2×2 determinant can be calculated as:

|C| = |a b c d|  = (a x d) – (b x c)

For example: C = |8 6 3 4|

Its determinant can be calculated as:

|C| = |8   6 3   4|

|C| = (8×4) – (6×3) = 32 – 18 = 14

Conclusion

Determinants are regarded as scaling factors in the context of matrices. In terms of the matrices themselves, they can be thought of as functions of stretching out and shrinking in respectively. When given a square matrix as input, determinants produce a single number as the result of the algorithm.