Operations on Determinants
Operations on Determinants:
- For any square matrix , .
- The determinant of a matrix and its transpose are equal.
Swapping rows or columns changes the sign of the determinant: , where is the matrix obtained after swapping rows or columns in .
Multiplying a row or column by a scalar multiplies the determinant by : for an matrix.
Adding a Multiple of One Row/Column to Another:
- If is obtained by adding a multiple of one row or column to another in , then .
- 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:
Inverses and Adjoint:
The determinant of a square matrix is nonzero if and only if is invertible.
The adjoint matrix is calculated using cofactors and transposes, which involves determinants.
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.
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.
Given two matrices and with determinants and :
- For Matrices:
For matrices :
The product of their determinants is:
- For Matrices:
For matrices :
The product of their determinants is not straightforward. It involves a more complex expansion of the matrices' elements according to the rule:
- For Higher-Dimensional Matrices:
Beyond matrices, calculating the product of determinants becomes increasingly intricate due to expansion methods and may involve computational complexity.
Let's take two matrices and multiply their determinants:
Calculating the determinants:
Now, let's find the product of the determinants:
Therefore, the product of the determinants of matrices and is .
Differentiation of Determinants
Given a square matrix of size with elements depending on a variable :
To differentiate the determinant of matrix with respect to (), we use the following formula:
- denotes the determinant of matrix .
- represents the trace of a matrix.
- is the inverse of matrix .
- represents the derivative of matrix with respect to .
Let's consider a matrix dependent on :
First, calculate the determinant of matrix :
Next, find the inverse of matrix :
Now, calculate , the derivative of matrix with respect to :
Next, find the trace of :
Finally, find using the formula:
This formula will yield the derivative of the determinant of matrix with respect to .
Summation of Determinants
Given two square matrices and of the same size , the sum or difference of their determinants can be found using the following rules:
Addition Rule: If and are matrices: The determinant of the sum of two matrices is the sum of their determinants.
Subtraction Rule: Similarly, if and are matrices: The 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.
Let's consider two matrices and :
First, calculate the determinants of matrices and :
Now, find the sum and difference of the determinants:
Sum of Determinants:
Difference of Determinants:
Thus, the sum of the determinants of matrices and is , and the difference of their determinants is .