Problem 15 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{rr}{7} & {5.2} \\ {-4} & {1.6}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 32.

Step 1: Identify the Elements of the Matrix

The given matrix is a 2x2 determinant:\[\begin{bmatrix} 7 & 5.2 \ -4 & 1.6\end{bmatrix}\]The elements of this matrix are: \(a = 7\), \(b = 5.2\), \(c = -4\), and \(d = 1.6\).

Step 2: Apply the Determinant Formula

For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated using the formula:\[\text{det} = ad - bc\]Substitute the values of \(a\), \(b\), \(c\), and \(d\) to find the determinant.

Step 3: Substitute the Values

Substitute the extracted values into the formula:\[\text{det} = (7)(1.6) - (5.2)(-4)\]

Step 4: Perform the Multiplications

Compute the products:\[7 \times 1.6 = 11.2\]\[5.2 \times (-4) = -20.8\]

Step 5: Calculate the Determinant

Insert the multiplication results into the determinant formula and perform the subtraction:\[\text{det} = 11.2 - (-20.8) = 11.2 + 20.8\]

Step 6: Simplify the Expression

Add the numbers to find the determinant:\[\text{det} = 32\]

Key Concepts

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices in linear algebra. It consists of two rows and two columns, which together form four elements. In a 2x2 matrix, each element has a specific position known by the combination of its row and column index. For example, in the matrix below:\[\begin{bmatrix}7 & 5.2 \-4 & 1.6\end{bmatrix}\]- The element '7' is at position (1,1)- '5.2' is at (1,2)- '-4' is at (2,1)- '1.6' is at (2,2)2x2 matrices are foundational in understanding more complex matrices and serve as the building blocks for many other concepts in linear algebra. They are often used in simulations and calculations that require manageable data sets.

Matrix Multiplication

Matrix multiplication is a way to combine two matrices to produce another matrix. It involves taking the rows of the first matrix and the columns of the second matrix and multiplying corresponding elements followed by summing up those products. It's essential to note that matrix multiplication is not as straightforward as multiplying individual numbers. It's governed by specific rules:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

For our 2x2 matrix, the process for finding its determinant involves semi-multiplication, where specific elements are multiplied and then combined to calculate a single scalar value - the determinant.

Determinant Formula

To find the determinant of a 2x2 matrix, you can use a simple formula. If we have a matrix \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]The determinant is calculated using the formula:\[det = ad - bc\]In this equation, you multiply the 'a' and 'd' (the elements on the main diagonal) and subtract the product of 'b' and 'c' (the elements on the other diagonal). For instance, with our matrix:\[a = 7, \ b = 5.2, \ c = -4, \ d = 1.6\]The determinant becomes:\[(7 \cdot 1.6) - (5.2 \cdot -4) = 11.2 + 20.8 = 32\]Using the determinant formula allows us to easily calculate this critical value, which provides insights into the properties of the matrix.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, matrices, and systems of linear equations. It is a powerful tool used in various scientific fields, including physics, computer science, and engineering. With matrices like the 2x2, linear algebra helps to perform operations such as:

• Determining solutions to systems of equations
• Transforming geometric data
• Exploring properties of spaces and transformations

Understanding fundamental concepts such as a matrix's determinant is vital because they allow us to: - Assess the invertibility of a matrix (a matrix with a non-zero determinant is invertible). - Interpret the scaling properties of linear transformations. As students advance in linear algebra, they will encounter increasingly complex structures and operations, but mastering the basics of determinants and matrix operations is a critical first step.