Question

If $$ A=\left|\begin{array}{ll} k & 3 \\ 3 & k \end{array}\right| $$ and \(\left|\mathrm{A}^{3}\right|=343\), then find the value of \(\mathrm{k}=\ldots\) \(\begin{array}{llll}\text { (a) } \pm 1 & \text { (b) } \pm 2 & \text { (c) } \pm 3 & \text { (d) } \pm 4\end{array}\)

Short answer

The value of k is \(k = \pm 4\).

Step-by-step solution

Find the determinant of A

To find the determinant of a 2x2 matrix, subtract the product of the diagonal elements from the product of the off-diagonal elements. In this case, the determinant is given by: \(Det(A) = k \times k - 3 \times 3\)

Raise the determinant to the power of 3

Since we are given that the determinant of A raised to the power of 3 is equal to 343, we can write the following equation: \((Det(A))^3 = 343\) Substitute the expression for the determinant of A from step 1: \((k^2 - 9)^3 = 343\)

Solve for k

To solve for k, take the cube root of both sides: \(k^2 - 9 = 7\) Now, we need to solve for k: \(k^2 = 16\) So, \(k = \pm 4\).

Choose the correct option

Since the possible values for k are ±4, the correct option is: (d) ±4

Key concepts

Matrix Algebra

Matrix Algebra involves various operations and properties associated with matrices, including addition, subtraction, multiplication, and determinant calculation. Matrices are fundamental in representing and solving linear equations efficiently. A matrix is typically a rectangular array of numbers arranged in rows and columns. In this context:

• A 2x2 matrix, like matrix A in the given exercise, has two rows and two columns, represented as: \[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]
• To perform basic arithmetic operations with matrices, they must have the same dimensions, except for multiplication, which is performed with unique conditions.
• Multiplying matrices can accommodate the transformation of systems and application of linear equations.

Understanding matrix algebra is essential for advancing to more complex topics like determinants, which summarise some matrix properties in a single number.

Determinant Properties

Determinants are special values calculated from a matrix, helping to understand various matrix properties such as singularity and invertibility. Determinants are a central concept in linear algebra and can be calculated for square matrices. Here is how determinants relate to different concepts:

• The determinant of a 2x2 matrix: This is calculated by multiplying its diagonal elements and then subtracting the product of its off-diagonal elements. For matrix A, the determinant is calculated as: \[\text{Det}(A) = (k \times k) - (3 \times 3) = k^2 - 9\]
• Determinant's properties: The determinant helps in understanding whether a matrix is invertible (non-zero determinant) or singular (zero determinant). For instance, if matrix A were singular, its determinant would equal zero.
• Exponentiation of determinants: Raising a determinant to a power involves calculating the determinant first, as shown in the solution, followed by raising that result to the power (3 in this exercise).

Mastering determinants enables you to explore deeper into the intricacies of matrix properties.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically expressed as \(ax^2 + bx + c = 0\). Solving for the unknown variable involves rearranging the equation so it equals zero and then using techniques like factoring, completing the square, or applying the quadratic formula. In our exercise, quadratic equations arise in solving for \(k\):

• The expression \(k^2 - 9 = 7\): Rearranges to form a standard quadratic equation \(k^2 = 16\).
• Solving for \(k\): Involves extracting square roots from both sides, which gives \(k = \pm 4\). This method is often used when the coefficient of \(x\) is zero in a quadratic equation.
• Quadratic solutions: Quadratics can have two real roots, one real root, or two complex roots, and understanding the nature of roots helps in determining the values satisfying the equation.

Being adept with quadratic equations allows you to tackle equations of this nature with confidence, revealing the possible solutions and their implications.