Problem 4 - Algebra and Trigonometry

Question

Evaluate each determinant. $$ \left|\begin{array}{rr} {7} & {9} \\ {-2} & {-5} \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given matrix is -17

Step 1: Identify Matrix Elements

The given matrix is \[ \left|\begin{array}{rr} {7} & {9} \ {-2} & {-5} \end{array}\right| \], where a is 7, b is 9, c is -2, and d is -5.

Step 2: Substitute values in formula

By substituting these values into the determinant formula, det(A) = ad - bc gives us det(A) = (7*(-5)) - (9*(-2)).

Step 3: Calculate

Proceeding with the calculation, we get det(A) = -35 - (-18), which upon further simplification gives det(A) = -17.

Key Concepts

Matrix Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, including whether the matrix is invertible and the volume distortion during the linear transformation the matrix represents.

For a 2x2 matrix, the determinant is calculated using a specific formula. If the matrix is represented by \[ \left|\begin{array}{cc} a & b \ c & d \end{array}\right| \], the determinant (denoted as det(A)) is computed as \( ad - bc \). This operation takes the product of the elements on the leading diagonal (a and d) and subtracts the product of the off-diagonal elements (b and c).

Understanding the calculation of a determinant is crucial in algebra, especially in solving systems of linear equations, finding eigenvalues, and in many applications across mathematics and engineering such as calculating the area or volume with respect to coordinate transformations.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is the unifying thread of almost all of mathematics and includes everything from solving simple equations to studying abstractions such as groups, rings, and fields.

When it comes to matrix algebra, one key operation is finding the determinant, which is often required to solve linear systems or to understand geometrical transformations. In our exercise, algebra is applied by substituting the values of a, b, c, and d into the determinant formula and simplifying the result. It's important to carry out these operations methodically to avoid sign errors, which are common pitfalls.

Substitution and Simplification in Algebra

For the given problem, we substitute 7 for a, 9 for b, -2 for c, and -5 for d. Then, we simplify the expression \( (7)(-5) - (9)(-2) \) through multiplication and subtraction, highlighting the use of basic algebraic techniques to find the determinant of a matrix.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a square matrix, consisting of 2 rows and 2 columns. Although it is simple, the 2x2 matrix is a building block for understanding more complicated matrix operations and plays a vital role in various mathematical concepts.

In the context of our problem, we dealt with a 2x2 matrix to calculate the determinant. The ease of calculation with a 2x2 matrix makes it a good starting point for students to grasp the concept of determinants before moving on to more complex matrices.

Practical Application of 2x2 Matrices

These small matrices are not only useful in educational settings but also in real-world applications, like encoding rotations in computer graphics, transforming between different coordinate systems in physics, or even in economics, where they can represent simple linear models.