Problem 40 - 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. $$(2 B)^{-1}(A B)^{T}$$

Step-by-Step Solution

Answer

The determinant of the given matrix is \(\frac{5}{16}\).

Step 1: Compute the determinant of (2B)

First, we need to find the determinant of the matrix (2B). Recall that when multiplying a matrix by a scalar, the determinant of the new matrix is the product of the scalar raised to the power of the matrix size (n) and the determinant of the original matrix: \[\operatorname{det}(cM) = c^n \operatorname{det}(M)\] Here, the matrix B has a size of 4x4, and the scalar is 2. So: \[\operatorname{det}(2B) = 2^4 \operatorname{det}(B) = 2^4 \cdot 3\]

Step 2: Compute determinant of (2B)^{-1}

Now, we should find the determinant of the inverse of (2B). Recall that the determinant of the inverse matrix is equal to the reciprocal of the determinant of the original matrix: \[\operatorname{det}(M^{-1}) = \frac{1}{\operatorname{det}(M)}\] Here, our original matrix is 2B, so: \[\operatorname{det}((2B)^{-1}) = \frac{1}{\operatorname{det}(2B)} = \frac{1}{2^4 \cdot 3}\]

Step 3: Compute determinant of (AB)

For the next step, we will find the determinant of the product AB. Recall the product rule for determinants: \[\operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B)\] Here, we were given det(A) = 5 and det(B) = 3. So: \[\operatorname{det}(AB) = 5 \cdot 3\]

Step 4: Compute the determinant of (AB)^T

Now, we need to find the determinant of the transpose of AB. Recall that the determinant of a transposed matrix is equal to the determinant of the original matrix: \[\operatorname{det}(M^T) = \operatorname{det}(M)\] Here, our original matrix is AB, so: \[\operatorname{det}((AB)^T) = \operatorname{det}(AB) = 5 \cdot 3\]

Step 5: Compute determinant of (2B)^{-1}(AB)^T

Finally, we will find the determinant of the product (2B)^{-1}(AB)^T. Using the determinant product rule again, \[\operatorname{det}((2B)^{-1}(AB)^T) = \operatorname{det}((2B)^{-1}) \cdot \operatorname{det}((AB)^T) = \frac{1}{2^4 \cdot 3} \cdot (5 \cdot 3)\] Calculating the final product, we have: \[ \operatorname{det}((2B)^{-1}(AB)^T) = \frac{5}{2^4} \] So the determinant of the given matrix is \(\frac{5}{16}\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra and involves combining two matrices to produce another matrix. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Each element of the resulting matrix is calculated by taking the dot product of the corresponding row of the first matrix and the corresponding column of the second matrix.
To illustrate this, let's consider two matrices, A and B:

• A is a matrix with dimensions \(m \times n\).
• B is a matrix with dimensions \(n \times p\).
• The product, AB, will be a matrix with dimensions \(m \times p\).

The operation involves complex calculations of sums and products, but it is a systematic way to combine two matrices. It's essential to follow the matrix multiplication rules carefully to ensure accurate results.

Transpose of a Matrix

The transpose of a matrix is an operation that flips a matrix over its diagonal, swapping the row and column indices of each element. When you transpose a matrix, the rows become columns and vice versa.
For example, if you have a matrix A:

• Row 1: [1, 2, 3]
• Row 2: [4, 5, 6]

The transpose of matrix A, denoted as \(A^T\), will be:

• Column 1: [1, 4]
• Column 2: [2, 5]
• Column 3: [3, 6]

The determinant of a transposed matrix stays the same as the original matrix. Therefore, \(\operatorname{det}(M^T) = \operatorname{det}(M)\). This property is particularly useful when dealing with equations and determinants involving transposed matrices.

Inverse of a Matrix

The inverse of a matrix A, denoted as \( A^{-1} \), is a matrix that, when multiplied by A, results in the identity matrix. Only square matrices (those with the same number of rows and columns) can have inverses. For a matrix to have an inverse, its determinant must be non-zero.
Here's how you can think about it:

• If \(A \cdot A^{-1} = I\), where I is the identity matrix, then \(A^{-1}\) is the inverse of A.
• The formula for finding an inverse involves the matrix's determinant: \( \operatorname{det}(A^{-1}) = \frac{1}{\operatorname{det}(A)} \).

Finding an inverse often involves complex calculations, but the concept is critical for solving systems of equations and other operations in linear algebra.

Scalar Multiplication of Matrices

Scalar multiplication involves multiplying each element of a matrix by a scalar (a single number). When a matrix A is multiplied by a scalar c, each entry of the matrix is scaled by this number.
This operation can be represented as:

• If A is a matrix, and c is a scalar, then \(cA\) is a new matrix where each element of A is multiplied by c.
• For a 4x4 matrix and a scalar value c: \(\operatorname{det}(cA) = c^4 \cdot \operatorname{det}(A)\).

Scalar multiplication is straightforward but very important when dealing with matrix operations, as it changes both the size and the value of the determinant with respect to the scalar's power related to the matrix's dimension.