Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}-0.3 & -0.1 & 0.9 \\\2.5 & 4.9 & -3.2 \\\\-0.1 & 0.4 & 0.8\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is \(-0.051\).

Step 1: Identify the Matrix

The given matrix is a 3x3 matrix: \[\begin{bmatrix}-0.3 & -0.1 & 0.9 \2.5 & 4.9 & -3.2 \-0.1 & 0.4 & 0.8\end{bmatrix}\]

Step 2: Use the Formula for 3x3 Determinant

To find the determinant of a 3x3 matrix, use the formula: \[ \operatorname{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]where the matrix elements are:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\]

Step 3: Substitute Values into the Formula

From the matrix:\(a = -0.3\), \(b = -0.1\), \(c = 0.9\)\(d = 2.5\), \(e = 4.9\), \(f = -3.2\)\(g = -0.1\), \(h = 0.4\), \(i = 0.8\)Substitute these into the determinant formula.

Step 4: Calculate Intermediate Products

Calculate the intermediate products:\( ei - fh = (4.9 \times 0.8) - (0.4 \times -3.2) = 3.92 + 1.28 = 5.2 \)\( di - fg = (2.5 \times 0.8) - (-3.2 \times -0.1) = 2 - 0.32 = 1.68 \)\( dh - eg = (2.5 \times 0.4) - (4.9 \times -0.1) = 1 + 0.49 = 1.49 \)

Step 5: Substitute Back into the Formula

Now substitute the intermediate results back into the determinant formula:\(\operatorname{det} = -0.3 \times 5.2 - (-0.1 \times 1.68) + 0.9 \times 1.49\)\(= -1.56 + 0.168 + 1.341\)

Step 6: Simplify the Expression to Find the Determinant

Add together the terms:\(-1.56 + 0.168 + 1.341 = -0.051\)Thus, the determinant of the matrix is \(-0.051\).

Key Concepts

3x3 Matrix

A 3x3 matrix is a collection of numbers arranged into three rows and three columns. Each position within this grid holds a number, which is also referred to as an element. Any matrix with this format is called a 3x3 matrix.

Understanding matrices can be a fundamental step towards addressing problems in mathematics and engineering. They align well with the way we use coordinates to describe things in the real world. In a matrix, rows run horizontally and columns run vertically:

• Rows: Represent horizontal sections of the matrix.
• Columns: Represent vertical sections of the matrix.

Indices represent the position of a number in the matrix, denoted generally as \( a_{ij} \), where \( i \) is the row number, and \( j \) is the column number. In a 3x3 matrix, these index numbers range from 1 to 3.

Matrix Algebra

Matrix algebra is a powerful tool in mathematics that extends algebraic concepts to matrices. Matrix algebra consists of operations such as addition, subtraction, and multiplication.

In the context of the 3x3 matrix determinant, the focus is on multiplication and how we can apply it to find a specific result. In matrix algebra, multiplying matrices involves combining rows and columns in a systematic way:

• Matrix Multiplication: This involves multiplying rows from the first matrix with columns from the second matrix and summing the products.
• Determinant Calculation: Specifically for a 3x3 matrix, this involves a particular formula that calculates a scalar value.

The determinant is particularly useful in matrix algebra as it helps in solving systems of linear equations, finding inverses, and characterizing matrix properties like singularity (when the determinant is zero, the matrix does not have an inverse).

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces (also called linear spaces), and linear transformations. It is a broad field that encompasses numerous subtopics. Determinants of matrices, including 3x3 matrices, form an essential part of linear algebra.

Linear algebra tools like determinants allow us to:

• Solve linear equations organized in matrix form, making it easier to manage large systems of equations.
• Interpret matrix properties, such as rank, which tells us about the solutions to the system.
• Transform geometric objects, an operation widely used in physics and engineering.

For example, understanding transformations via matrices can help us determine how objects rotate or scale. Linear algebra provides the foundation for these operations.