In linear algebra, the determinant of a square matrix is a scalar value that encapsulates key properties of the matrix, including its invertibility and the volume scaling effect it has on geometric objects. One of the fundamental properties of determinants is that the determinant of the product of two matrices is equal to the product of their determinants. Formally, this property states that for any square matrices A and B of the same order, we have det(AB) = det(A) det(B). This result, known as the determinant product property, holds immense significance in various areas of mathematics, physics, and engineering, particularly in solving systems of linear equations, analyzing transformations, and calculating volumes. This article will explore different methods to prove this important theorem, delving into the underlying concepts and showcasing its practical applications.
Proof by Induction on the Order of Matrix B
One elegant way to prove the determinant product property is by employing mathematical induction on the order of matrix B.
Base Case: When B is a 1×1 matrix, the result holds trivially. Since det(B) is simply the single entry of B, and AB is obtained by multiplying each row of A by this single entry, det(AB) is equal to the product of the entries in A multiplied by the single entry in B, which is precisely det(A) det(B).
Inductive Step: Assume that the result holds for all matrices B of order k. We need to show that it holds for matrices B of order k+1. Consider a (k+1)×(k+1) matrix B. We can express B in block matrix form:
B = [ B11 B12 ]
[ B21 B22 ]
where B11 is a k×k matrix, B12 is a k×1 column vector, B21 is a 1×k row vector, and B22 is a scalar.
Now, consider the product AB:
AB = [ A B11 A B12 ]
[ A B21 A B22 ]
We can expand the determinant of AB using the cofactor expansion along the last row:
det(AB) = A B21 * det([A B12]) - A B22 * det([A B11])
By the induction hypothesis, det([A B11]) = det(A) det(B11) and det([A B12]) = det(A) det(B12). Substituting these into the above equation, we get:
det(AB) = A B21 * det(A) det(B12) - A B22 * det(A) det(B11)
= det(A) (A B21 * det(B12) - A B22 * det(B11))
= det(A) det(B)
This completes the inductive step, establishing that the result holds for all matrices B of order k+1. Therefore, by the principle of mathematical induction, det(AB) = det(A) det(B) holds for all square matrices A and B.
Proof using Row Operations and the Effect on Determinants
Another approach to prove the determinant product property relies on the understanding that elementary row operations on a matrix have predictable effects on its determinant.
Elementary Row Operations:
- Swapping two rows: Multiplies the determinant by -1.
- Multiplying a row by a scalar k: Multiplies the determinant by k.
- Adding a multiple of one row to another row: Does not change the determinant.
Proof:
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Reduce B to upper triangular form: By applying a sequence of elementary row operations to B, we can transform it into an upper triangular matrix U. Each elementary row operation on B will correspond to an equivalent operation on AB. Therefore, if we denote the sequence of operations that transform B to U as R, we have:
R(B) = U R(AB) = AU -
Determinant of upper triangular matrices: The determinant of an upper triangular matrix is simply the product of its diagonal elements.
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Effect of row operations on determinants: Let det(R) represent the combined effect of all elementary row operations on the determinant. Using the properties of elementary row operations outlined above, we can write:
det(R(B)) = det(R) det(B) det(R(AB)) = det(R) det(AB) -
Equating determinants: Since R(B) = U and R(AB) = AU, we have:
det(U) = det(R) det(B) det(AU) = det(R) det(AB) -
Applying the determinant of an upper triangular matrix:
det(U) = det(A) det(U) det(AU) = det(A) det(U) -
Conclusion: Since det(U) ≠ 0 (assuming B is invertible), we can divide both sides by det(U), leading to the conclusion det(AB) = det(A) det(B).
Implications and Applications
The determinant product property has far-reaching implications and applications in various mathematical and scientific disciplines. Here are some notable examples:
1. Solving Systems of Linear Equations:
The determinant product property plays a crucial role in solving systems of linear equations using Cramer's rule. Cramer's rule provides a direct solution for the unknowns in a system of equations based on determinants. It utilizes the determinant product property to relate the determinant of the coefficient matrix to the determinants of matrices obtained by replacing specific columns with the constant vector of the system.
2. Linear Transformations and Volume Scaling:
The determinant of a matrix representing a linear transformation provides information about the scaling factor of the transformation in terms of volume. The determinant product property shows that the overall scaling factor for a composition of two linear transformations is the product of their individual scaling factors, consistent with the intuitive notion of volume scaling.
3. Eigenvalues and Eigenvectors:
The determinant product property is essential in determining the eigenvalues of a matrix. Eigenvalues represent the scaling factors associated with the eigenvectors of a linear transformation, and they are found by solving the characteristic equation, which involves determinants. The determinant product property facilitates simplifying the characteristic equation and solving for the eigenvalues.
4. Invertibility and Rank:
The determinant of a matrix directly relates to its invertibility and rank. A matrix is invertible if and only if its determinant is nonzero. The rank of a matrix is the number of linearly independent rows or columns, and it is also related to the determinant by the fact that a matrix has full rank if and only if its determinant is nonzero. The determinant product property helps in determining the invertibility and rank of products of matrices.
5. Matrix Inverses:
The determinant product property is instrumental in deriving the formula for the inverse of a matrix. The inverse of a matrix exists if and only if its determinant is nonzero. The formula for the inverse involves the adjoint matrix, which is calculated using cofactors, and the determinant. The determinant product property ensures that the inverse of a product of matrices is the product of their inverses in reverse order.
Conclusion
The determinant product property det(AB) = det(A) det(B) is a fundamental theorem in linear algebra with numerous implications and applications across various disciplines. It simplifies calculations involving determinants, provides valuable insights into the behavior of linear transformations, and facilitates the solution of systems of linear equations. By understanding and appreciating this property, we gain a deeper understanding of the properties and applications of determinants in linear algebra.
