Question

If \(\mathrm{D}_{1}=\left|\begin{array}{ccc}\mathrm{x} & \mathrm{a} & \mathrm{a} \\\ \mathrm{a} & \mathrm{x} & \mathrm{a} \\ \mathrm{a} & \mathrm{a} & \mathrm{x}\end{array}\right|\) and \(\mathrm{D}_{2}=\left|\begin{array}{ll}\mathrm{x} & \mathrm{a} \\ \mathrm{a} & \mathrm{x}\end{array}\right|\), then... (a) \(D_{1}=3\left(\mathrm{D}_{2}\right)^{(3 / 2)}\) (b) \(D_{1}=3 \mathrm{D}_{2}^{2}\) (c) \((\mathrm{d} / \mathrm{dx})\left(\mathrm{D}_{1}\right)=3 \mathrm{D}_{2}^{2}\) (d) \((\mathrm{d} / \mathrm{dx})\left(\mathrm{D}_{1}\right)=3 \mathrm{D}_{2}\)

Short answer

The determinants \(D_1\) and \(D_2\) are calculated as \(D_1 = x^3 - 2 * a^3\) and \(D_2 = x^2 - a^2\), respectively. The correct relationship between \(D_1\) and \(D_2\) is given by option (d): \(\frac{d}{dx}(D_1) = 3 * D_2\). This is derived by first finding the derivative of \(D_1\) with respect to \(x\) and proving that it is equal to \(3 * D_2\).

Step-by-step solution

Calculate D1

To find the determinant D1 of a 3x3 matrix, we use the following formula: D1 = x * (x * x - a * a) - a * (a * x - a * a) + a * (a * a - a * x) Now let's simplify: D1 = x^3 - 2 * a^3

Calculate D2

To find the determinant D2 of a 2x2 matrix, we use the following formula: D2 = x * x - a * a Now let's simplify: D2 = x^2 - a^2 Now, let's analyze the given options and find the correct answer.

Check Option (a)

Check if D1 = 3*(D2)^(3/2): D1 = x^3 - 2 * a^3 D2 = x^2 - a^2 3 * (D2)^(3/2) = 3 * [(x^2 - a^2)^(3/2)] This doesn't match the expression for D1, so option (a) is incorrect.

Check Option (b)

Check if D1 = 3 * D2^2: D1 = x^3 - 2 * a^3 D2 = x^2 - a^2 3 * D2^2 = 3 * (x^2 - a^2)^2 This doesn't match the expression for D1, so option (b) is incorrect.

Check Option (c)

Check if (d/dx)(D1) = 3 * D2^2: First, find the derivative of D1 with respect to x: (d/dx)(D1) = (d/dx)(x^3 - 2 * a^3) = 3 * x^2 Now, check if 3 * x^2 = 3 * (x^2 - a^2)^2: 3 * x^2 ≠ 3 * (x^2 - a^2)^2 This doesn't match, so option (c) is incorrect.

Check Option (d)

Check if (d/dx)(D1) = 3 * D2: First, find the derivative of D1 with respect to x as we did before: (d/dx)(D1) = (d/dx)(x^3 - 2 * a^3) = 3 * x^2 Now, check if 3 * x^2 = 3 * (x^2 - a^2): 3 * x^2 = 3 * (x^2 - a^2) + 3 * a^2 This matches, so option (d) is correct. The correct answer is (d) (d/dx)(D1) = 3 * D2.

Key concepts

Determinants

Determinants play a vital role in matrix algebra, serving as a stepping stone towards more complex concepts such as eigenvalues and matrix inverses. They are scalar values that can be computed from the elements of a square matrix. For instance, the determinant of a 2x2 matrix, denoted as \( \text{D}_2 \), is calculated by subtracting the product of its diagonals, which can be formally represented as \( \text{D}_2 = x^2 - a^2 \).

When dealing with a 3x3 matrix, the process gets a bit intricate, involving combinations of the 2x2 minors and their respective cofactors. The determinant of such a matrix, \( \text{D}_1 \), is obtained through a specific formula leading to \( \text{D}_1 = x^3 - 2a^3 \). The fundamental property that the determinant of a matrix with at least two identical rows or columns is zero is not seen here, but it's essential in simplifying calculations in more complex matrices. Understanding these calculations is critical for assessing matrix solvability and behavior in systems of linear equations.

Matrix Algebra

Matrix algebra is an extension of traditional algebra with sets of numbers, called matrices, encapsulating complex data in a structured form. It comprises operations such as addition, subtraction, multiplication, and inversion that adhere to specific rules distinct from real number arithmetic. The exercises we encounter often require proficiency in these operations to deduce properties like the determinant.

For example, to accurately find the relationship between \( \text{D}_1 \) and \( \text{D}_2 \) in the problem, one must not only perform the operations within each determinant but also understand how these operations interrelate. The ability to decompose a 3x3 determinant into smaller 2x2 determinants simplifies the process and illustrates the interconnected nature of these mathematical objects. When faced with more complex operations, such as raising a determinant to a power or taking its derivative, a solid grasp on matrix algebra becomes indispensable.

Derivative Calculus

Derivative calculus is concerned with the change of quantities; it is the mathematical study of continuous change and is instrumental in various scientific fields. In our current context, we apply derivatives to the function that represents the determinant of matrices with respect to a variable. The derivative of \( \text{D}_1 \) with respect to \( x \) is denoted as \( (\frac{d}{dx})(\text{D}_1) \), leading to \( 3x^2 \) after applying the power rule of derivative calculus to \( x^3 \).

The solution process involves checking against different expressions for \( \text{D}_2 \) and its derived forms, confirming whether or not these expressions are equivalent to \( (\frac{d}{dx})(\text{D}_1) \). The final part of our solution illustrates the application of derivative calculus to determine that option (d) \( (\frac{d}{dx})(\text{D}_1) = 3 * \text{D}_2 \) is correct. This ties back to the fundamental concept of derivatives being the rate of change - in this case, the rate at which the determinant of a 3x3 matrix changes concerning one of its variable entries.