Solving For K Determinant Of 5P - 2Q - 3I Equals 298

In the captivating realm of linear algebra, matrices reign supreme as fundamental entities that elegantly encapsulate and manipulate numerical data. Within this domain, the determinant emerges as a powerful scalar invariant, meticulously summarizing the essence of a square matrix and unveiling its intrinsic properties. This article embarks on an in-depth exploration of a captivating problem involving matrices P and Q, intertwined by the intriguing concept of determinants. Our mission is to unravel the mystery surrounding the determinant of a specific matrix expression, $5P - 2Q - 3I$, where P and Q are given matrices, and I represents the revered identity matrix. By meticulously dissecting the problem, we will not only decipher the solution but also illuminate the underlying mathematical principles that govern the behavior of matrices and determinants.

Keywords: matrices, determinant, linear algebra, matrix expression, identity matrix, scalar invariant, matrix properties, mathematical principles

Let us consider two matrices, P and Q, defined as follows:

$$P = \begin{pmatrix} 1 & k \ 3 & 4 \end{pmatrix}$$

$$Q = \begin{pmatrix} -3 & 0 \ 1 & -2 \end{pmatrix}$$

where k is an unknown scalar that we aim to determine. Our quest is further guided by the knowledge that the determinant of the matrix expression $5P - 2Q - 3I$ equals 298, where I is the ubiquitous 2 x 2 identity matrix, given by:

$$I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$

Our mission is to embark on a mathematical journey to unravel the value of k, the elusive scalar that intricately binds the matrices P and Q within this determinant equation.

To embark on our quest to determine the value of k, we shall adopt a strategic approach, meticulously dissecting the problem into manageable steps. Our roadmap involves the following key milestones:

1. Construct the Matrix Expression: We shall begin by meticulously constructing the matrix expression $5P - 2Q - 3I$, substituting the given matrices P, Q, and I into the equation.
2. Calculate the Determinant: With the matrix expression in hand, we shall compute its determinant, employing the fundamental rules of determinant calculation for 2 x 2 matrices.
3. Formulate the Equation: The determinant, once calculated, will unveil an equation involving the unknown scalar k. We shall carefully formulate this equation, setting it up for our grand finale.
4. Solve for k: Finally, we shall employ our algebraic prowess to solve the equation for k, unveiling the value that satisfies the given condition.

Step 1: Construct the Matrix Expression

Let us embark on our journey by constructing the matrix expression $5P - 2Q - 3I$. Substituting the given matrices P, Q, and I, we obtain:

$$5P - 2Q - 3I = 5\begin{pmatrix} 1 & k \ 3 & 4 \end{pmatrix} - 2\begin{pmatrix} -3 & 0 \ 1 & -2 \end{pmatrix} - 3\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$

Now, we perform scalar multiplication and matrix subtraction, meticulously combining the corresponding elements:

$$5P - 2Q - 3I = \begin{pmatrix} 5 & 5k \ 15 & 20 \end{pmatrix} - \begin{pmatrix} -6 & 0 \ 2 & -4 \end{pmatrix} - \begin{pmatrix} 3 & 0 \ 0 & 3 \end{pmatrix}$$

$$5P - 2Q - 3I = \begin{pmatrix} 5 - (-6) - 3 & 5k - 0 - 0 \ 15 - 2 - 0 & 20 - (-4) - 3 \ \end{pmatrix}$$

$$5P - 2Q - 3I = \begin{pmatrix} 8 & 5k \ 13 & 21 \end{pmatrix}$$

Thus, we have successfully constructed the matrix expression $5P - 2Q - 3I$.

Step 2: Calculate the Determinant

Our next step is to calculate the determinant of the matrix expression we just constructed. For a 2 x 2 matrix of the form $\begin{pmatrix} a & b \ c & d \end{pmatrix}$, the determinant is given by $ad - bc$. Applying this formula to our matrix, we get:

$$det(5P - 2Q - 3I) = det(\begin{pmatrix} 8 & 5k \ 13 & 21 \end{pmatrix}) = (8)(21) - (5k)(13)$$

$$det(5P - 2Q - 3I) = 168 - 65k$$

We have now successfully calculated the determinant of the matrix expression.

Step 3: Formulate the Equation

Our problem statement provides us with the crucial piece of information that the determinant of $5P - 2Q - 3I$ equals 298. Equating our calculated determinant to this value, we formulate the equation:

$$168 - 65k = 298$$

This equation now stands as the gateway to unveiling the value of k.

Step 4: Solve for k

To solve for k, we shall employ our algebraic skills. Rearranging the equation, we get:

$$-65k = 298 - 168$$

$$-65k = 130$$

Dividing both sides by -65, we obtain:

$$k = \frac{130}{-65}$$

$$k = -2$$

Thus, we have successfully unveiled the value of k, the elusive scalar that satisfies the given condition.

In this captivating exploration, we have successfully navigated the realm of matrices, determinants, and linear algebra to determine the value of k. By meticulously constructing the matrix expression, calculating its determinant, formulating the equation, and solving for k, we have unveiled that k = -2. This journey has not only provided us with a numerical solution but has also illuminated the power of determinants as scalar invariants that encapsulate the essence of matrices. We have witnessed how determinants can be manipulated and utilized to solve intricate problems involving matrices, showcasing their fundamental role in the mathematical landscape.

Summary of Key Steps:

• Constructed the matrix expression $5P - 2Q - 3I$.
• Calculated the determinant of the matrix expression.
• Formulated the equation by equating the determinant to 298.
• Solved the equation for k, obtaining k = -2.

This problem exemplifies the intricate interplay between matrices, determinants, and linear algebra, underscoring the elegance and power of mathematical tools in unraveling complex problems.

For those who crave further exploration in the realm of matrices and determinants, the following avenues beckon:

• Explore higher-order determinants: Delve into the computation of determinants for matrices of sizes 3 x 3 and beyond, uncovering the patterns and techniques involved.
• Investigate determinant properties: Unravel the fascinating properties of determinants, such as their behavior under matrix operations and their relationship to matrix invertibility.
• Applications of determinants: Discover the diverse applications of determinants in various fields, including solving linear systems, calculating eigenvalues, and determining matrix rank.

By venturing into these avenues, you shall further deepen your understanding of the captivating world of matrices and determinants, unlocking their potential to solve a multitude of mathematical puzzles and real-world problems.

Matrices and determinants form the bedrock of linear algebra, serving as powerful tools for representing and manipulating numerical data. Within this framework, the determinant emerges as a crucial scalar invariant, encapsulating the essence of a square matrix and revealing its intrinsic properties. This article delves into a fascinating problem involving matrices P and Q, interconnected through the concept of determinants. Our objective is to unravel the mystery surrounding the determinant of the expression $5P - 2Q - 3I$, where P and Q are given matrices, and I represents the identity matrix. Through a meticulous step-by-step approach, we aim to decipher the solution while highlighting the fundamental mathematical principles governing matrices and determinants.

Keywords: Linear Algebra, Matrices, Determinants, Matrix Operations, Scalar Invariant, Identity Matrix

Consider two matrices P and Q, defined as:

$$P = \begin{pmatrix} 1 & k \ 3 & 4 \end{pmatrix}$$

$$Q = \begin{pmatrix} -3 & 0 \ 1 & -2 \end{pmatrix}$$

where k is an unknown scalar. Given that the determinant of the matrix $5P - 2Q - 3I$ equals 298, where I is the 2 x 2 identity matrix:

$$I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$

our goal is to determine the value of the unknown scalar k.

To achieve our goal of finding k, we will adopt a structured approach:

1. Matrix Expression Construction: First, we construct the matrix $5P - 2Q - 3I$ by substituting the given matrices P, Q, and I.
2. Determinant Calculation: Next, we calculate the determinant of the resulting matrix, applying the rules of determinant calculation for 2 x 2 matrices.
3. Equation Formulation: The determinant calculation will yield an equation involving k. We will formulate this equation.
4. Solving for k: Finally, we solve the equation to find the value of k that satisfies the given condition.

Step 1: Matrix Expression Construction

We begin by substituting P, Q, and I into the expression $5P - 2Q - 3I$:

$$5P - 2Q - 3I = 5 \begin{pmatrix} 1 & k \ 3 & 4 \end{pmatrix} - 2 \begin{pmatrix} -3 & 0 \ 1 & -2 \end{pmatrix} - 3 \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$

Performing scalar multiplication and matrix subtraction:

$$5P - 2Q - 3I = \begin{pmatrix} 5 & 5k \ 15 & 20 \end{pmatrix} - \begin{pmatrix} -6 & 0 \ 2 & -4 \end{pmatrix} - \begin{pmatrix} 3 & 0 \ 0 & 3 \end{pmatrix}$$

Combining corresponding elements:

$$5P - 2Q - 3I = \begin{pmatrix} 5 - (-6) - 3 & 5k - 0 - 0 \ 15 - 2 - 0 & 20 - (-4) - 3 \ \end{pmatrix} = \begin{pmatrix} 8 & 5k \ 13 & 21 \end{pmatrix}$$

Thus, we have successfully constructed the matrix expression $5P - 2Q - 3I$.

Step 2: Determinant Calculation

For a 2 x 2 matrix $\begin{pmatrix} a & b \ c & d \end{pmatrix}$, the determinant is given by $ad - bc$. Applying this to our matrix:

$$det(5P - 2Q - 3I) = det(\begin{pmatrix} 8 & 5k \ 13 & 21 \end{pmatrix}) = (8)(21) - (5k)(13)$$

$$det(5P - 2Q - 3I) = 168 - 65k$$

We have now calculated the determinant of the matrix expression.

Step 3: Equation Formulation

We are given that $det(5P - 2Q - 3I) = 298$. Therefore, we set our calculated determinant equal to 298:

$$168 - 65k = 298$$

This equation represents the core of our problem.

Step 4: Solving for k

Rearranging the equation:

$$-65k = 298 - 168$$

$$-65k = 130$$

Dividing both sides by -65:

$$k = \frac{130}{-65}$$

$$k = -2$$

Therefore, the value of k is -2.

Through a systematic approach, we successfully determined the value of k to be -2. This solution was achieved by constructing the matrix expression, calculating its determinant, formulating an equation, and solving for k. This exercise demonstrates the application of determinant properties in solving problems involving matrices.

This exploration highlights the interconnectedness of matrices, determinants, and linear algebra principles. By understanding these concepts, we can effectively solve complex mathematical problems.

Key Steps Recap:

• Constructed the matrix expression $5P - 2Q - 3I$.
• Calculated the determinant: $168 - 65k$.
• Formulated the equation: $168 - 65k = 298$.
• Solved for k: $k = -2$.

This problem illustrates the elegance and utility of determinants in matrix algebra.

To expand your knowledge of matrices and determinants, consider the following:

• Higher-Order Determinants: Investigate methods for computing determinants of 3 x 3 and larger matrices.
• Determinant Properties: Explore properties such as how row operations affect determinants and the relationship between determinants and matrix invertibility.
• Applications: Learn about the applications of determinants in solving linear equations, finding eigenvalues, and determining matrix rank.

By pursuing these areas, you will deepen your understanding of matrices and determinants, enhancing your problem-solving skills in mathematics and related fields. The problem we addressed serves as a foundation for more advanced topics in linear algebra, demonstrating the critical role determinants play in the broader mathematical landscape.