Operations on Determinants

Operations on Determinants:

  1. Transpose Property:

    • For any square matrix A, det(A)=det(AT).
    • The determinant of a matrix and its transpose are equal.
  2. Row/Column Operations:

    • Swapping rows or columns changes the sign of the determinant: det(A)=det(A), where A is the matrix obtained after swapping rows or columns in A.

    • Multiplying a row or column by a scalar k multiplies the determinant by k: det(kA)=kn×det(A) for an n×n matrix.

  3. Adding a Multiple of One Row/Column to Another:

    • If A is obtained by adding a multiple of one row or column to another in A, then det(A)=det(A).
  4. Row/Column Expansion:

    • Expansion of a determinant along a row or column yields the sum of determinants of smaller matrices obtained by excluding that row or column.

Application in Matrix Operations:

  1. Inverses and Adjoint:

    • The determinant of a square matrix A is nonzero if and only if A is invertible.

    • The adjoint matrix Adj(A) is calculated using cofactors and transposes, which involves determinants.

  2. Solving Systems of Equations:

    • Determinants play a crucial role in determining the solvability of systems of linear equations.

    • Cramer's rule uses determinants to express the solutions of a system in terms of determinants of smaller matrices.

  3. Geometry and Transformations:

    • The absolute value of the determinant of a matrix representing a linear transformation determines the scaling factor and orientation change in space.

Multiplying Determinants

Given two matrices A and B with determinants A and B:

  1. For 2×2Matrices:

For matrices 2×2:

A=[abcd] B=[efgh]

The product of their determinants is: A×B=(adbc)×(ehfg)

  1. For 3×3 Matrices:

For matrices 3×3:

A=[abcdefghi] B=[pqrstuvwx]

The product of their determinants is not straightforward. It involves a more complex expansion of the matrices' elements according to the rule:

A×B=abcdefghi×pqrstuvwx

  1. For Higher-Dimensional Matrices:

Beyond 3×3 matrices, calculating the product of determinants becomes increasingly intricate due to expansion methods and may involve computational complexity.

Example:

Let's take two 2×2 matrices and multiply their determinants:

A=[2314] B=[5120]

Calculating the determinants:

A=(2×4)(3×(1))=8+3=11

B=(5×0)(1×2)=02=2

Now, let's find the product of the determinants:

A×B=11×(2)=22

Therefore, the product of the determinants of matrices A and B is 22.

Differentiation of Determinants

Given a square matrix A of size n×n with elements depending on a variable x:

A(x)=[a11(x)a12(x)a1n(x)a21(x)a22(x)a2n(x)an1(x)an2(x)ann(x)]

To differentiate the determinant of matrix A with respect to x (ddxA), we use the following formula:

ddxA=ATr(A1dAdx)

Where:

  • Adenotes the determinant of matrix A.
  • Tr represents the trace of a matrix.
  • A1 is the inverse of matrix A.
  • dAdx represents the derivative of matrix A with respect to x.

Example:

Let's consider a 2×2 matrix A(x) dependent on x:

A(x)=[xx23x2]

First, calculate the determinant of matrix A(x):

A=x×2x2×3x=2x3x3

Next, find the inverse of matrix A(x):

A1(x)=12x3x3[2x23xx]

Now, calculate dAdx, the derivative of matrix A(x) with respect to x:

dAdx=[12x30]

Next, find the trace of A1dAdx:

A1dAdx=12x3x3[2x23xx][12x30]

 A1dAdx=12x3x3[23x22x33x33x]

Finally, find ddxA using the formula:

ddxA=(2x3x3)Tr(12x3x3[23x22x33x33x])

This formula will yield the derivative of the determinant of matrix A(x) with respect to x.

Summation of Determinants

Given two square matrices A and B of the same size n×n, the sum or difference of their determinants can be found using the following rules:

  1. Addition Rule: If A and B are n×n matrices: A+B=A+B The determinant of the sum of two matrices is the sum of their determinants.

  2. Subtraction Rule: Similarly, if A and B are n×n matrices: AB=ABThe determinant of the difference of two matrices is the difference of their determinants.

These rules apply when the matrices involved are of the same size.

Example:

Let's consider two 3×3 matrices A and B:

A=[213401235] B=[120321142]

First, calculate the determinants of matrices A and B:

A=213401235=(2×0×5)+(1×(1)×2)+(3×4×3)(3×0×2)(2×(1)×2)(5×4×1)=18

B=120321142=(1×2×2)+(2×1×(1))+(0×3×4)(0×2×(1))(2×3×(1))(1×1×4)=8

Now, find the sum and difference of the determinants:

  1. Sum of Determinants: A+B=A+B=18+8=26

  2. Difference of Determinants: AB=AB=188=10

Thus, the sum of the determinants of matrices A and B is 26, and the difference of their determinants is 10.