Problem 44 - Differential Equations and Linear Algebra

Question

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$(4 B)^{3}$$

Step-by-Step Solution

Answer

The determinant of the given matrix is \(4^{12} * 27\).

Step 1: Rewrite the given matrix

Rewrite (4B)^3 as (4^3)(B^3).

Step 2: Apply the properties of determinants

Using property 1, we know that det(4^3) = 4^(3*4). Now using property 2, we know that det(B^3) = det(B)^3.

Step 3: Compute the determinant

We are given det(B) = 3. Therefore, det(B^3) = 3^3 = 27. Then compute det(4^3), which is 4^12. Now we can find the determinant of the given matrix: det((4B)^3) = det(4^3 * B^3) = det(4^3) * det(B^3) = 4^12 * 27. Thus, the determinant of the given matrix is \(4^{12} * 27\).

Key Concepts

Properties of Determinants

Determinants are fascinating tools in linear algebra. They are special numbers associated with a square matrix. Understanding their properties helps solve complex matrix problems effectively.

Here are some key properties of determinants:

• The determinant of a product of two matrices is equal to the product of their determinants. Mathematically, this is expressed as \( \det(AB) = \det(A) \cdot \det(B) \).
• For a scalar multiple of a matrix, if \( c \) is a scalar and \( A \) is an \( n \times n \) matrix, then \( \det(cA) = c^n \cdot \det(A) \).
• The determinant of a power of a matrix, \( A^k \), is \( (\det(A))^k \).
• Swapping two rows or columns in a matrix changes the sign of the determinant.

Applying these properties makes calculating determinants much simpler, especially for larger matrices.

In the context of our exercise, these properties help simplify the computation of \( \det((4B)^3) \) by breaking it down into multipliers we can easily calculate.

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. It involves multiplying the rows of the first matrix by the columns of the second matrix.

Understanding how to perform matrix multiplication is vital for solving problems involving determinants. Here's a quick overview of the rules:

• The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.
• To find the element in the resulting matrix at position \((i, j)\), multiply each element of the \(i\)-th row of the first matrix by the corresponding element of the \(j\)-th column of the second matrix, and sum the results.
• The resulting matrix will have dimensions based on the rows of the first matrix and columns of the second matrix.

In our problem, we specifically use properties of determinants with matrix multiplication. For example, the property \( \det(AB) = \det(A) \cdot \det(B) \) heavily relies on understanding these multiplications.

Determinant Computation

Computing determinants involves applying various properties and formulas to simplify calculations. When dealing with large matrices or powers of matrices, this can be especially handy.

For a matrix raised to a power, like \( B^3 \), we can use the properties of determinants to simplify the computation:

• Start by finding \( \det(B) \). In our problem, \( \det(B) = 3 \).
• Apply the property \( \det(B^k) = (\det(B))^k \) to find \( \det(B^3) \), which equals \( 3^3 = 27 \).

Next, for scalar multiples, consider \((4B)^3\). The determinant of a scalar multiple \( cA \) is \( c^n \cdot \det(A) \), where \( n \) is the matrix dimension. Here, \( \det(4^3) = 4^{3 \times 4} = 4^{12} \).

Finally, combine these to find \( \det((4B)^3) \):

• Multiply the determinants: \( \det(4^3 \cdot B^3) = \det(4^3) \cdot \det(B^3) = 4^{12} \cdot 27 \).

This structured approach to determinant computation ensures accuracy and simplifies potentially tedious calculations.