Problem 16 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{rr}{-3.2} & {-5.8} \\ {4.1} & {3.9}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 11.3.

Step 1: Understanding the Determinant of a 2x2 Matrix

For any 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated using the formula \( ad - bc \). This operation combines the products of the diagonals.

Step 2: Assign Values from the Matrix

From the given matrix \( \begin{pmatrix} -3.2 & -5.8 \ 4.1 & 3.9 \end{pmatrix} \), assign the values: \( a = -3.2 \), \( b = -5.8 \), \( c = 4.1 \), and \( d = 3.9 \).

Step 3: Calculate the Product of the First Diagonal

Calculate the product of the first diagonal \( ad = (-3.2)(3.9) \). This gives \( -12.48 \).

Step 4: Calculate the Product of the Second Diagonal

Calculate the product of the second diagonal \( bc = (-5.8)(4.1) \). This gives \( -23.78 \).

Step 5: Subtract the Second Diagonal Product from the First

Apply the determinant formula: subtract the product of the second diagonal from the first diagonal product: \( ad - bc = -12.48 - (-23.78) = -12.48 + 23.78 \).

Step 6: Compute the Resulting Value

Perform the arithmetic operation to find \( -12.48 + 23.78 = 11.3 \).

Step 7: Conclusion

The determinant of the given matrix is \( 11.3 \).

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple form of a square matrix characterized by having two rows and two columns. In general, these matrices can be represented as follows:

• The top row of the matrix is labeled as \( a \) and \( b \).
• The bottom row of the matrix is labeled as \( c \) and \( d \).

This makes a 2x2 matrix look like this:\[\begin{pmatrix}a & b \c & d \end{pmatrix}\]The simplicity of the 2x2 matrix is one of the reasons it is often used in basic matrix operations.
It is a great starting point for anyone who is new to matrix theory and calculations. However, even within this simplicity, powerful operations like determinant calculation can be performed.

Matrix Operations

Matrix operations allow us to perform a variety of calculations on matrices, utilizing their rows and columns effectively. Some fundamental operations you can perform on matrices include:

• Addition and Subtraction: Adding or subtracting matrices involves combining or removing values that are in similar positions within two matrices.
• Scalar Multiplication: This operation involves multiplying every element of a matrix by a fixed number, known as a scalar.
• Matrix Multiplication: Unlike scalar multiplication, this involves arrays of the corresponding elements in two matrices to create a new matrix. For 2x2 matrices, each element in the resulting matrix is the sum of products.
• Determinant Calculation: This is a special operation that provides a scalar value representing certain properties of a square matrix.

These operations can help in applications such as solving systems of equations, transforming geometric data, and more. Understanding these basics paves the way to mastering more complex matrix operations.

Determinant Calculation

The determinant is a valuable scalar associated with square matrices. For a 2x2 matrix, the determinant is calculated using a straightforward formula. The determinant of matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is determined by the following expression:\[ det(A) = ad - bc \]This calculation involves two key steps:

• Computing the product of the elements on the main diagonal of the matrix, which are \( a \) and \( d \).
• Subtracting the product of the elements on the other diagonal, which are \( b \) and \( c \).

In essence, the determinant is the difference between these two computed values. It is important because it offers insights into the properties of matrices, such as whether a matrix is invertible. A determinant of zero indicates the matrix does not have an inverse, which is important in various mathematical applications like solving linear systems.