Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rr}5 & x \\\\-3 & 2\end{array}\right]=6$$

Step-by-Step Solution

Answer

The value of \(x\) is \(-\frac{4}{3}\).

Step 1: Define the Determinant Formula for 2x2 Matrix

Recall that the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \).

Step 2: Substitute Matrix Elements into the Formula

Given the matrix \( \begin{bmatrix} 5 & x \ -3 & 2 \end{bmatrix} \), identify \(a = 5\), \(b = x\), \(c = -3\), and \(d = 2\), and plug these values into the determinant formula: \( 5 \times 2 - (-3) \times x \).

Step 3: Simplify the Determinant Expression

Simplify the expression from Step 2 as follows: \(5 \times 2 - (-3)x = 10 + 3x\).

Step 4: Set the Determinant Equal to 6

Since \(\det\) of the matrix is given as 6, set the equation equal to 6: \[10 + 3x = 6\].

Step 5: Solve for x

Solve the equation \(10 + 3x = 6\) by first subtracting 10 from both sides, giving \(3x = -4\), then divide both sides by 3 to isolate \(x\): \[x = -\frac{4}{3}\].

Key Concepts

2x2 matrix

A 2x2 matrix is a simple yet fundamental concept in matrix algebra. It consists of two rows and two columns, forming a square matrix. In essence, a 2x2 matrix is written as: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}.\]This structure is small but powerful, allowing us to perform various mathematical operations, including the calculation of determinants.
The determinant of a 2x2 matrix is critical as it assists in solving equations and finding inverses. To find the determinant, we use the formula \(ad - bc\), where \(a, b, c,\) and \(d\) are the elements of the matrix.

• Determinants help determine if a matrix has an inverse.
• If the determinant is zero, the matrix does not have an inverse, making it singular.

Understanding this concept is essential for further explorations in linear algebra.

matrix algebra

Matrix algebra facilitates the manipulation and understanding of matrices, which are arrays of numbers. This branch of mathematics extends operations of addition, subtraction, multiplication, and finding determinants to these matrices.
Determinant of a Matrix
A determinant is a crucial value computed from a square matrix, indicating various properties. For a 2x2 matrix, it is calculated using the formula: \(ad - bc\).

• This value tells us if transformations represented by the matrix are reversible.
• It influences the matrix's behavior in systems of linear equations, impacting the solutions' existence and uniqueness.

Understanding matrix algebra allows students to solve complex systems more simply and efficiently.

problem solving in mathematics

Problem solving in mathematics involves a strategic approach to finding solutions to mathematical questions. It goes beyond simple calculations, requiring logical thinking and applying relevant mathematical principles.
The Importance of Step-by-Step Solutions
Solving equations, such as finding the determinant of a matrix, reflects this structured problem-solving approach.

• Break the problem into manageable steps, such as defining the determinant's formula for the 2x2 matrix.
• Substitute known values systematically to simplify the expressions.
• Carefully solve the resulting equation for the unknown variable.

This method not only simplifies calculations but also ensures all aspects of the problem are addressed, enhancing understanding and accuracy.