Product of Two Determinants

In linear algebra, the determinant is a scalar value that is a function of a square matrix. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation described by the matrix. When dealing with two square matrices, it is often useful to understand how the determinants of these matrices interact, especially when considering the product of the matrices.

Properties of Determinants

Before we delve into the product of two determinants, let's review some important properties of determinants:

• Multiplicative Property: The determinant of the product of two matrices is equal to the product of their determinants.

$$ \det(AB) = \det(A) \cdot \det(B) $$

• Invertible Matrix: A matrix is invertible if and only if its determinant is non-zero.

• Transpose: The determinant of a matrix and its transpose are equal.

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

• Scalar Multiplication: If a matrix is multiplied by a scalar, the determinant is multiplied by the scalar raised to the power of the matrix size (n).

$$ \det(kA) = k^n \cdot \det(A) $$

Product of Two Determinants

When considering the product of two determinants, we can use the multiplicative property of determinants. This property states that for any two square matrices ( A ) and ( B ) of the same size, the determinant of their product is equal to the product of their determinants.

Formula

The formula for the product of two determinants is straightforward:

$$ \det(AB) = \det(A) \cdot \det(B) $$

Examples

Let's consider two ( 2 \times 2 ) matrices ( A ) and ( B ):

$$ A = \begin{bmatrix} a & b \ c & d \end{bmatrix}, \quad B = \begin{bmatrix} e & f \ g & h \end{bmatrix} $$

The determinants of ( A ) and ( B ) are:

$$ \det(A) = ad - bc, \quad \det(B) = eh - fg $$

According to the multiplicative property, the determinant of the product of ( A ) and ( B ) is:

$$ \det(AB) = (ad - bc)(eh - fg) $$

Table of Differences and Important Points

Property	Description	Relevance to Product of Determinants
Multiplicative Property	The determinant of the product of two matrices equals the product of their determinants.	Fundamental property for the product of two determinants.
Invertibility	A matrix is invertible if its determinant is non-zero.	If either matrix has a determinant of zero, the product matrix is not invertible.
Transpose	The determinant of a matrix is equal to the determinant of its transpose.	The product property holds for transposes as well.
Scalar Multiplication	Multiplying a matrix by a scalar affects the determinant.	When considering the product of matrices, scalar multiplication can change the determinant product.

Conclusion

Understanding the product of two determinants is crucial for various applications in linear algebra. The multiplicative property simplifies the computation of determinants for the product of matrices and is essential for solving systems of linear equations, finding eigenvalues, and more. Remember that this property only holds for square matrices and is one of the many interesting properties that make determinants a powerful tool in mathematics.