Problem 43 - 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. $$B^{-1}(2 A) B^{T}$$

Step-by-Step Solution

Answer

The determinant of the given matrix \(B^{-1}(2 A) B^{T}\) is \(80\).

Step 1: Identify determinant properties related to the given matrix

We have the given matrix as \(B^{-1}(2 A) B^{T}\). We need to use the properties of determinants mentioned in the analysis section to break down this product and find its determinant.

Step 2: Apply property 1 - product of determinants

Using property 1, we can write the determinant of the product matrix as: \(\operatorname{det}(B^{-1}(2 A) B^{T}) = \operatorname{det}(B^{-1}) \operatorname{det}(2A) \operatorname{det}(B^{T})\)

Step 3: Apply properties 3, 4, and 5

Now, we can apply the other properties of determinants to simplify the expression further: - Property 5: \(\operatorname{det}(B^{-1}) = \frac{1}{\operatorname{det}(B)} = \frac{1}{3}\) - Property 3: \(\operatorname{det}(2A) = 2^n \operatorname{det}(A) = 2^4 \cdot 5 = 80\) - Property 4: \(\operatorname{det}(B^{T}) = \operatorname{det}(B) = 3\)

Step 4: Multiply the simplified determinants

Now that we have simplified each determinant in the product, we can multiply them together: \(\operatorname{det}(B^{-1}(2 A) B^{T}) = \frac{1}{3} \cdot 80 \cdot 3 = 80\) So, the determinant of the given matrix is \(80\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra. It involves two matrices, let's call them matrix \(A\) and matrix \(B\), combined to form a new matrix. However, multiplying matrices is not as straightforward as multiplying numbers. Here’s a simple way to understand it:

• The number of columns in the first matrix \(A\) must be equal to the number of rows in the second matrix \(B\).
• The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.

The element in the \(i^{th}\) row and \(j^{th}\) column of the product matrix is calculated by multiplying the elements of the \(i^{th}\) row of the first matrix by the corresponding elements of the \(j^{th}\) column of the second matrix, and then summing these products. Each step may look complex, but practice makes it easier! To perform matrix multiplication smoothly, ensure that the matrices are compatible and follow a methodical approach.

Inverse Matrix

An inverse matrix is like a magical undo button for matrices. If you have matrix \(A\), the inverse of \(A\), denoted as \(A^{-1}\), is such that when you multiply \(A\) by \(A^{-1}\), you get the identity matrix: \[ AA^{-1} = A^{-1}A = I \]

• The identity matrix \(I\) is the version of 1 for matrices. It's a square matrix with ones on the diagonal and zeros elsewhere.
• For a matrix to have an inverse, it must be square, meaning it has the same number of rows and columns, and its determinant must not be zero.

Finding an inverse involves a few mathematical operations and is not always possible for every matrix. In many applications, particularly in solving linear equations and transformations, having an inverse is crucial, hence the importance of understanding how to work with inverse matrices.

Transpose Matrix

The transpose of a matrix flips it over its diagonal. If you have a matrix \(A\), its transpose is denoted as \(A^T\). This transformation does the following:

• Switches rows with columns.
• The \(i^{th}\) row becomes the \(i^{th}\) column and vice versa.

Taking a transpose is particularly useful in operations like finding symmetric matrices where \(A = A^T\), or simplifying multiplication processes such as finding the determinant of a transpose matrix, which is equal to the original matrix’s determinant. The concept of transposing may seem simple, but it holds significant importance in matrix operations and various applications across mathematics and sciences.

Properties of Determinants

Determinants are special numbers that can be associated with a square matrix. Here are some key properties:

• The determinant of the identity matrix is always 1.
• If a matrix is multiplied by a scalar, its determinant is multiplied by the scalar raised to the number of rows (or columns). For example, \( \det(kA) = k^n \det(A) \) where \(n\) is the size of the square matrix.
• The determinant of the inverse matrix is the reciprocal of the determinant of the original matrix, \(\det(A^{-1}) = 1/\det(A)\).
• The determinant of a transpose matrix is equal to the determinant of the original matrix, i.e., \(\det(A^T) = \det(A)\).

These properties make determinants highly useful in determining matrix invertibility and solving linear equations. They help break down complex matrices into more manageable calculations, allowing us to deduce many properties about the solution.