Problem 30 - Differential Equations and Linear Algebra

Question

A matrix that is a scalar multiple of \(I_{n}\) is called an \(n \times n\) scalar matrix. (a) Determine the \(4 \times 4\) scalar matrix whose trace is 8 (b) Determine the \(3 \times 3\) scalar matrix such that the product of the elements on the main diagonal is 343

Step-by-Step Solution

Answer

\( (a) \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{bmatrix}, (b) \begin{bmatrix} 7 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 7 \end{bmatrix} \)

Step 1: (a) Step 1: Find the trace of a 4x4 scalar matrix

The trace of a matrix is the sum of the elements on the main diagonal. For a 4x4 scalar matrix, the trace can be represented as: Trace = k + k + k + k Since we know the given trace is 8, we can set up the equation: 8 = k + k + k + k

Step 2: (a) Step 2: Solve for k in 4x4 scalar matrix

We can solve for k by simplifying the equation: 8 = 4k Divide both sides of the equation by 4: k = 2

Step 3: (a) Step 3: Construct the 4x4 scalar matrix

Now that we have found k, we can construct the 4x4 scalar matrix: \( \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{bmatrix} \)

Step 4: (b) Step 1: Find the product of the main diagonal for a 3x3 scalar matrix

The product of the main diagonal elements in a 3x3 scalar matrix can be represented as: Product = k * k * k Since we are given that the product of the elements on the main diagonal is 343, we can set up the equation: 343 = k^3

Step 5: (b) Step 2: Solve for k in 3x3 scalar matrix

To solve for k, we can take the cube root of both sides: k = \(\sqrt[3]{343}\) k = 7

Step 6: (b) Step 3: Construct the 3x3 scalar matrix

Now that we have found k, we can construct the 3x3 scalar matrix: \( \begin{bmatrix} 7 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 7 \end{bmatrix} \)

Key Concepts

Trace of a Matrix

When it comes to understanding matrix operations, the concept of the trace of a matrix is crucial. In a nutshell, the trace is the sum of all the values along the main diagonal of a square matrix. The main diagonal is the line of entries that extends from the top left corner to the bottom right corner of the matrix.

For example, if we have a matrix
\[ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \]

then the trace of A, denoted as Tr(A), is the sum
\[ \text{Tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} \].

In scalar matrices, this concept simplifies since all the non-diagonal entries are zero and every diagonal entry is the same scalar value, denoted as k. So, for a scalar matrix, it's simply the scalar value multiplied by the number of rows (or columns, since it's square). This characteristic makes calculations involving the trace of a scalar matrix particularly straightforward, as seen in the given textbook exercise.

Main Diagonal of a Matrix

Moving beyond the trace, let's discuss the main diagonal of a matrix. This is not just a part of the definition of the trace, but a structural feature of any square matrix. The main diagonal is essentially the backbone of a matrix. For any square matrix with the form
\[ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \],

the main diagonal will be the set of elements a_{11}, a_{22}, ..., a_{nn}. This diagonal is significant in matrix algebra as it influences determinant calculations, the trace, and properties of specific matrices like identity matrices and scalar matrices.

For scalar matrices used in the textbook exercise, the main diagonal is distinctive because every element on it is the same, representing the scalar multiple.

Calculation in Scalar Matrices

In our exercise, the calculation of the main diagonal was a point of focus for determining the entire matrix. For example, the product of the elements on this diagonal was given for a 3x3 scalar matrix. Since each element is the same scalar k, the product is simply k raised to the power of the number of elements in the main diagonal, or k^3 for a 3x3 matrix.

Matrix Algebra

Diving into the broader topic of matrix algebra, we encounter rules and operations that dictate how matrices are added, subtracted, multiplied, and more. Matrix algebra is important as it forms the foundation for solving systems of linear equations, transforming geometric figures, and various applications in engineering, physics, and computer science.

In the context of our exercise, we apply matrix algebra to construct scalar matrices and to determine their properties such as the trace and the main diagonal.

Importance of Scalar Matrices

Scalar matrices are particularly interesting because they are multiples of the identity matrix, which itself is a fundamental entity in matrix algebra. The identity matrix acts like 1 in regular multiplication—it doesn't change the matrix it is multiplied with.

• In a scalar matrix, since all off-diagonal entries are zero, operations like multiplying two scalar matrices result in another scalar matrix of the same size.
• Understanding properties of scalar matrices helps in simplifying matrix equations and in finding more complex matrix properties like inverses and eigenvalues.

The exercise solutions applied these principles to solve for the unknown scalar values, illustrating how matrix algebra is not just about performing operations, but also about understanding the underlying structure and properties of matrices.