Problem 6 - Algebra and Trigonometry

Question

Evaluate each determinant.Therefore, the given determinant evaluates to -22 $$ \left|\begin{array}{rr} {1} & {-3} \\ {-8} & {2} \end{array}\right| $$

Step-by-Step Solution

Answer

-22

Step 1: Identify the elements

In the 2x2 matrix, the elements are: \(a=1\), \(b=-3\), \(c=-8\), and \(d=2\). The method for finding the determinant of a 2x2 matrix is \(ad-bc\).

Step 2: Substitute the elements into the formula

The next step is to substitute the values into the formula. This will look like this: (1*2)-(-3*-8).

Step 3: Simplify the equation

By multiplying and subtracting, the equation simplifies to: 2-24.

Step 4: Solve the equation

The calculation becomes: -22. Therefore, the determinant of the given matrix is -22.

Key Concepts

Determinant of a Matrix

The determinant is a special value that can be calculated from a square matrix. It is a single number that provides important information about the matrix, such as whether the matrix is invertible and the volume factor of linear transformations.

In the case of a 2x2 matrix, calculating the determinant is straightforward. It involves the difference of the products of its diagonals. If you are presented with a matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \], the determinant of matrix A, denoted as \( |A| \) or \( \det(A) \) is found by the formula \( ad - bc \).

This is a crucial concept in linear algebra, as the determinant is used to solve systems of linear equations, find the inverse of a matrix, and in various areas of advanced mathematics and physics.

2x2 Matrix

A 2x2 matrix is a square array of numbers that consists of two rows and two columns. It is often represented in the form \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] where \( a, b, c, \) and \( d \) are elements of the matrix. The organization of these elements is essential, as their position directly affects the value of the determinant.

The simplicity of the 2x2 matrix makes it a perfect introductory concept for understanding how more complex matrices behave. When dealing with determinants in a 2x2 matrix, it is vital to remember the significance of the positions: \(a \) and \(d \) are diagonal counterparts, as are \( b \) and \( c \). This arrangement is the foundation for the determinant formula for a 2x2 matrix.

Algebraic Techniques

Algebraic techniques are the methods used to manipulate and solve algebraic expressions and equations. These techniques can include operations such as addition, subtraction, multiplication, and division, as well as more complex actions like factoring, expanding, and simplifying expressions.

When evaluating the determinant of a 2x2 matrix, for example, you apply arithmetic operations to multiple the elements \( a \) and \( d \) and then \( b \) and \( c \), followed by the subtraction of their products. Simplifying this resulting expression is crucial to arrive at the correct determinant.

Importance of Sign

It's important to take note of the signs of the elements when calculating the determinant. For instance, if an element is negative, as in our example \( b = -3 \) and \( c = -8 \), their product will be positive \( (-3) \times (-8) = 24 \), which significantly affects the outcome. Understanding how to handle negative numbers is essential in determining the correct determinant value.