Let A And B Be Square Matrices Of Order 4 Such That |A| = -5 And |B| = 3. Find: A) |2AB| B) |adj(AB)| C) |5A^{-1}B^T| D) |A^TB^{-1}A^2|

In the realm of linear algebra, determinants play a crucial role in understanding the properties of matrices and their transformations. Determinants, being scalar values computed from square matrices, provide significant insights into matrix invertibility, the volume scaling factor of linear transformations, and the solutions of linear systems. This article delves into calculating determinants of various matrix operations involving two 4x4 matrices, A and B, given their individual determinants. We will explore the properties of determinants under scalar multiplication, matrix multiplication, adjoint operations, inverses, and transposes. By understanding these properties, we can efficiently compute complex expressions involving determinants, solidifying our grasp on linear algebra principles.

Given two square matrices, A and B, both of order 4, we know that the determinant of A, denoted as |A|, is -5, and the determinant of B, denoted as |B|, is 3. Our objective is to find the determinants of several matrix operations: |2AB|, |adj(AB)|, |5A⁻¹Bᵀ|, and |AᵀB⁻¹A²|. Each of these expressions involves different matrix operations and properties of determinants, providing a comprehensive exercise in determinant calculations. In the subsequent sections, we will meticulously break down each calculation, providing detailed explanations and leveraging the fundamental properties of determinants to arrive at the solutions. This exploration will not only enhance our computational skills but also deepen our understanding of the theoretical underpinnings of matrix algebra.

The first task is to find the determinant of the matrix 2AB. To accomplish this, we will use the properties of determinants concerning scalar multiplication and matrix multiplication. First, let's consider the scalar multiplication property. If A is an n x n matrix and c is a scalar, then |cA| = cⁿ|A|. In our case, A and B are 4x4 matrices, so n = 4. Applying this property, we know that |2AB| involves multiplying the matrix AB by the scalar 2. Thus, we will have a factor of 2⁴.

Next, we consider the determinant of a product of matrices. The determinant of the product of two matrices is the product of their determinants, that is, |AB| = |A||B|. This property is fundamental and allows us to separate the determinants of the individual matrices. Now, we can combine these two properties to find |2AB|. We have |2AB| = 2⁴|AB| because multiplying a 4x4 matrix by a scalar means each of the 4 rows is multiplied by that scalar. Then, using the property of the determinant of a product, we get |2AB| = 2⁴|A||B|. We are given that |A| = -5 and |B| = 3. Plugging these values into the equation, we have |2AB| = 2⁴(-5)(3). Calculating this, 2⁴ is 16, so |2AB| = 16 * (-5) * 3 = 16 * (-15) = -240. Therefore, the determinant of 2AB is -240.

Now, we need to calculate the determinant of the adjugate of the matrix AB, denoted as |adj(AB)|. The adjugate of a matrix, also known as the adjoint, is the transpose of the cofactor matrix. A crucial property that relates a matrix to its adjugate is: A * adj(A) = adj(A) * A = |A|I, where I is the identity matrix. From this property, we can derive the relationship between the determinant of the adjugate and the determinant of the original matrix. Specifically, for an n x n matrix A, |adj(A)| = |A|ⁿ⁻¹. In our case, we are dealing with 4x4 matrices, so n = 4.

First, we find the determinant of the product AB using the property |AB| = |A||B|. We know |A| = -5 and |B| = 3, so |AB| = (-5)(3) = -15. Now, we need to find |adj(AB)|. Using the formula |adj(A)| = |A|ⁿ⁻¹, we replace A with AB, and since AB is a 4x4 matrix, n = 4. Therefore, |adj(AB)| = |AB|⁴⁻¹ = |AB|³. We found that |AB| = -15, so we have |adj(AB)| = (-15)³. Calculating this gives us (-15)³ = -15 * -15 * -15 = -3375. Hence, the determinant of the adjugate of AB is -3375. This result highlights the importance of understanding the properties of adjugates and how they relate to the determinant of the original matrix, providing a powerful tool for matrix analysis.

In this section, we aim to find the determinant of the matrix 5A⁻¹Bᵀ. This involves several key properties of determinants, including those related to scalar multiplication, inverses, and transposes. First, let’s address the scalar multiplication. Recall that for an n x n matrix A and a scalar c, |cA| = cⁿ|A|. In our case, we have the scalar 5 multiplying the matrix A⁻¹Bᵀ, and since these are 4x4 matrices, n = 4. Thus, the scalar multiplication will contribute a factor of 5⁴.

Next, we consider the inverse of a matrix. The determinant of the inverse of a matrix A is the reciprocal of the determinant of A, i.e., |A⁻¹| = 1/|A|. We are given that |A| = -5, so |A⁻¹| = 1/(-5) = -1/5. Now, let’s look at the transpose. The determinant of the transpose of a matrix is the same as the determinant of the original matrix, i.e., |Bᵀ| = |B|. We are given that |B| = 3, so |Bᵀ| = 3. Now, we can combine all these properties to calculate |5A⁻¹Bᵀ|.

We have |5A⁻¹Bᵀ| = 5⁴|A⁻¹Bᵀ|. Using the property that the determinant of a product is the product of the determinants, we get |5A⁻¹Bᵀ| = 5⁴|A⁻¹||Bᵀ|. Now, substituting the individual determinants, we have |5A⁻¹Bᵀ| = 5⁴ * (-1/5) * 3. Calculating this gives us 5⁴ = 625, so |5A⁻¹Bᵀ| = 625 * (-1/5) * 3 = 625 * (-3/5) = -375. Therefore, the determinant of 5A⁻¹Bᵀ is -375. This calculation underscores the significance of understanding how different matrix operations affect the determinant and the importance of applying the appropriate properties.

Finally, we need to calculate the determinant of the matrix AᵀB⁻¹A². This expression involves the transpose, inverse, and powers of matrices. To compute this determinant, we will use the properties that relate determinants to these matrix operations. We already know that the determinant of the transpose of a matrix is equal to the determinant of the matrix itself, i.e., |Aᵀ| = |A|. We are given that |A| = -5, so |Aᵀ| = -5. We also know that the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix, i.e., |B⁻¹| = 1/|B|. Since |B| = 3, we have |B⁻¹| = 1/3.

Now, let's consider A². This is simply A multiplied by itself, so A² = A * A. Using the property that the determinant of a product is the product of the determinants, we have |A²| = |A * A| = |A||A| = |A|². We know |A| = -5, so |A²| = (-5)² = 25. Now, we can combine these results to find |AᵀB⁻¹A²|. Using the property that the determinant of a product is the product of the determinants, we have |AᵀB⁻¹A²| = |Aᵀ||B⁻¹||A²|. Substituting the determinants we found, we get |AᵀB⁻¹A²| = (-5) * (1/3) * 25.

Calculating this gives us |AᵀB⁻¹A²| = -5 * (1/3) * 25 = -125/3. Therefore, the determinant of AᵀB⁻¹A² is -125/3. This comprehensive calculation demonstrates how the properties of determinants can be combined to solve complex problems involving various matrix operations. Understanding these properties is crucial for efficient and accurate matrix analysis in linear algebra.

In this exploration, we have successfully calculated the determinants of several matrix operations involving matrices A and B. By applying key properties of determinants, such as those related to scalar multiplication, matrix multiplication, adjugates, inverses, and transposes, we were able to efficiently compute the values of |2AB|, |adj(AB)|, |5A⁻¹Bᵀ|, and |AᵀB⁻¹A²|. These calculations not only reinforced our understanding of determinant properties but also highlighted their practical applications in linear algebra.

The determinant of 2AB was found to be -240, which involved using the properties of scalar multiplication and matrix multiplication. The determinant of the adjugate of AB, |adj(AB)|, was calculated as -3375, demonstrating the relationship between a matrix's determinant and its adjugate's determinant. For the determinant of 5A⁻¹Bᵀ, we found the value to be -375, utilizing the properties of scalar multiplication, inverses, and transposes. Finally, the determinant of AᵀB⁻¹A² was computed as -125/3, showcasing the combined application of transpose, inverse, and matrix multiplication properties. These detailed calculations serve as a valuable exercise in mastering determinant computations and their underlying principles.

Through these examples, it is evident that a strong grasp of determinant properties is essential for tackling complex matrix operations. The ability to manipulate and simplify expressions involving determinants is crucial in various fields, including engineering, physics, and computer science, where matrices are fundamental tools. By understanding and applying these principles, one can gain deeper insights into the behavior and characteristics of matrices and linear transformations. This article serves as a testament to the power and elegance of linear algebra in solving intricate problems, emphasizing the importance of these fundamental concepts in mathematical and scientific endeavors.