Problem 23 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{rr} -9 & -1 \\ -10 & -5 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 35.

Step 1: Identifying Elements

In the given 2x2 determinant, identify the elements as follows: \(a = -9\), \(b = -1\), \(c = -10\), and \(d = -5\).

Step 2: Apply Determinant Formula

For a 2x2 matrix \(\left| \begin{array}{cc} a & b \ c & d \end{array} \right|\), the determinant is calculated using the formula \(ad - bc\).

Step 3: Substitute Elements into Formula

Substitute the identified elements into the formula: \((-9)(-5) - (-1)(-10)\).

Step 4: Calculate Products

Calculate each product individually: \((-9)(-5) = 45\) and \((-1)(-10) = 10\).

Step 5: Compute the Difference

Subtract the product of \(bc\) from the product of \(ad\): \(45 - 10 = 35\).

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix you will encounter in mathematics. A matrix is essentially a rectangular array of numbers or variables. With a 2x2 matrix, you have two rows and two columns.

Essentially, a 2x2 matrix looks like this:

• The first row: two elements (let's call them 'a' and 'b')
• The second row: two elements (let's call them 'c' and 'd')

So, when we write down a 2x2 matrix, it can be represented as:
\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]
This format helps you locate and use the individual elements within a matrix when performing operations such as finding a determinant. The beauty of the 2x2 matrix lies in its simplicity and ease of computation, making it an excellent starting point for learning about matrices.

The Determinant Formula for a 2x2 Matrix

The determinant is a special number that can be calculated from a square matrix (like a 2x2). It has a lot of applications, including solving systems of linear equations and finding the inverse of a matrix.

To compute the determinant of a 2x2 matrix, we use a straightforward formula:

• Given the matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
• The determinant is calculated as: \( ad - bc \)

Here’s how you use this formula:- Multiply the top-left element (\(a\)) with the bottom-right element (\(d\)).- Multiply the top-right element (\(b\)) with the bottom-left element (\(c\)).- Subtract the second product from the first.

In the context of the exercise, substituting the values into the formula lets us find the determinant easily and helps us interpret the results in meaningful ways, such as determining if the matrix is invertible.

Exploring Matrices

A matrix is a fundamental tool in mathematics, particularly in areas such as algebra, calculus, and applied mathematics. A matrix is simply a grid of numbers arranged into rows and columns. These numbers can represent various physical quantities, transformations, or even relationships in systems of equations.

Working with matrices involves numerous operations:

• Matrix addition and subtraction, where matrices of the same dimension are combined element-wise.
• Scalar multiplication involves multiplying each element of a matrix by a scalar value.
• Matrix multiplication, a more complex process where the rows of the first matrix are multiplied by the columns of the second matrix, requiring careful attention to dimensions.

Understanding these operations gives you deeper mathematical insights and enhances your problem-solving skills.

Matrices are powerful because they enable you to organize and manipulate sets of data in a compact form. Plus, they are a cornerstone in solving systems of linear equations, optimizing linear programming problems, and performing rotations and translations in computer graphics.