Reflection Property in Determinants

The reflection property in the context of determinants refers to the property that the determinant of a matrix remains unchanged when its rows and columns are interchanged. In other words, the determinant of a matrix is equal to the determinant of its transpose.

Definition

For any square matrix $A$, the reflection property can be stated as:

$$ \det(A) = \det(A^T) $$

where $A^T$ is the transpose of matrix $A$.

Explanation

The transpose of a matrix is obtained by swapping the rows and columns. The reflection property tells us that this operation does not affect the value of the determinant. This is a fundamental property of determinants and is true for matrices of any size.

Table of Differences and Important Points

Aspect	Determinant of Matrix $A$	Determinant of Transpose $A^T$
Definition	$\det(A)$ is the scalar value that represents a certain volume scaling factor and orientation associated with the linear transformation described by $A$.	$\det(A^T)$ is the determinant of the matrix obtained by reflecting $A$ over its main diagonal.
Reflection Property	$\det(A)$ remains unchanged when rows and columns are interchanged.	$\det(A^T)$ is equal to $\det(A)$ due to the reflection property.
Computation	Determinant is computed using various methods such as expansion by minors, row reduction, or using the Leibniz or Laplace formula.	Determinant is computed in the same way as for $A$, since $A^T$ is just a rearrangement of $A$.
Effect of Transposition	Transposing the matrix does not affect the determinant.	Reflecting the matrix does not introduce any change in the determinant value.

Formulas

The determinant of a matrix can be calculated using various methods. For a 2x2 matrix:

$$ \det(A) = \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc $$

For a 3x3 matrix using the Sarrus rule:

$$ \det(A) = \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = aei + bfg + cdh - ceg - bdi - afh $$

Examples

Example 1: 2x2 Matrix

Let's consider a 2x2 matrix and its transpose:

$$ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \quad A^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix} $$

The determinant of $A$ is:

$$ \det(A) = (1)(4) - (2)(3) = 4 - 6 = -2 $$

The determinant of $A^T$ is:

$$ \det(A^T) = (1)(4) - (3)(2) = 4 - 6 = -2 $$

As we can see, $\det(A) = \det(A^T)$, which confirms the reflection property.

Example 2: 3x3 Matrix

Consider a 3x3 matrix and its transpose:

$$ B = \begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 1 & 0 & 6 \end{bmatrix}, \quad B^T = \begin{bmatrix} 1 & 0 & 1 \ 2 & 4 & 0 \ 3 & 5 & 6 \end{bmatrix} $$

The determinant of $B$ is:

$$ \det(B) = (1)(4)(6) + (2)(5)(1) + (3)(0)(0) - (3)(4)(1) - (2)(0)(6) - (1)(5)(0) = 24 + 10 + 0 - 12 - 0 - 0 = 22 $$

The determinant of $B^T$ is:

$$ \det(B^T) = (1)(4)(6) + (0)(0)(3) + (1)(2)(5) - (1)(4)(3) - (0)(2)(6) - (1)(0)(5) = 24 + 0 + 10 - 12 - 0 - 0 = 22 $$

Again, $\det(B) = \det(B^T)$, which confirms the reflection property for a 3x3 matrix.

Conclusion

The reflection property is an essential characteristic of determinants that simplifies many computations and proofs in linear algebra. It is particularly useful when dealing with transposed matrices, as it guarantees that the determinant of a matrix and its transpose are equal. This property is independent of the size of the matrix and holds for all square matrices.