Problem 17 - A Graphical Approach to College Algebra

Question

Find each determinant. $$\operatorname{det}\left[\begin{array}{ccc}10 & 2 & 1 \\\\-1 & 4 & 3 \\\\-3 & 8 & 10\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 166.

Step 1: Understand the Matrix

We are given a 3x3 matrix, and our task is to find its determinant. The matrix is:\[\begin{bmatrix}10 & 2 & 1 \-1 & 4 & 3 \-3 & 8 & 10\end{bmatrix}\] The determinant of a 3x3 matrix will be calculated by applying the formula involving the elements of the matrix.

Step 2: Apply the Determinant Formula

The determinant \( \det(A) \) for a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is given by:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]For our given matrix, identify: \( a = 10, b = 2, c = 1, d = -1, e = 4, f = 3, g = -3, h = 8, i = 10 \).

Step 3: Calculate Each Cofactor

Let's break down the calculation into individual cofactor components:1. Compute \( ei - fh \): \[ ei - fh = (4 \times 10) - (3 \times 8) = 40 - 24 = 16 \]2. Compute \( di - fg \): \[ di - fg = (-1 \times 10) - (3 \times -3) = -10 + 9 = -1 \]3. Compute \( dh - eg \): \[ dh - eg = (-1 \times 8) - (4 \times -3) = -8 + 12 = 4 \]

Step 4: Substitute Cofactor Values

Substitute the cofactor results back into the determinant formula:\[\det(A) = 10(16) - 2(-1) + 1(4)\]This simplifies to:\[\det(A) = 160 + 2 + 4\]

Step 5: Calculate the Final Determinant

Add up all the terms from the previous step:\[160 + 2 + 4 = 166\] Therefore, the determinant of the given matrix is 166.

Key Concepts

3x3 Matrix

A 3x3 matrix refers to a mathematical array consisting of three rows and three columns. Each element in the matrix is a number or expression which is organized into rows and columns, like grid patterns. An example of a 3x3 matrix is:\[\begin{bmatrix}10 & 2 & 1 \-1 & 4 & 3 \-3 & 8 & 10\end{bmatrix}\]3x3 matrices are a fundamental component in linear algebra. They are often used to represent systems of equations, perform transformations in geometry, and much more. The arrangement allows for various operations and calculations, such as determining the determinant, an important value in understanding the properties of the matrix.

Cofactor Expansion

Cofactor expansion is a technique used to calculate the determinant of a matrix, especially for matrices larger than 2x2 like our 3x3 matrix. It involves the use of minors and cofactors. A 'minor' is the determinant of a smaller matrix, formed by removing one row and one column from the original matrix, while a 'cofactor' is a minor signed based on its position.### Steps Involved in Cofactor Expansion:- Choose any row or column from the matrix to perform cofactor expansion. It's often strategic to choose the row or column with the most zeros to simplify calculations.- Calculate the determinant of the 2x2 minors corresponding to the selected elements.- Multiply each minor by its respective cofactor, which involves an alternating sign pattern.An example calculation using cofactor expansion for our matrix involves using the first row,\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]where each term involves calculating the determinant of a 2x2 matrix from elements not in the same row or column as the cofactor.

Matrix Algebra

Matrix Algebra involves mathematical operations applied to matrices and encompasses concepts like addition, multiplication, inverse, and determinant calculation. It is an extension of regular algebra to accommodate the structure and properties of matrices. ### Key Operations in Matrix Algebra:

• Addition: Add corresponding elements from two matrices of the same size.
• Multiplication: Multiply rows by columns, which leads to a new matrix. This can also provide transformations in geometry.
• Determination of Determinant: Determines scalar values that tell us about the system's properties, like invertibility.

Understanding determinants, like in our exercise, is crucial in applications such as solving linear equations, where determining if a unique solution exists depends on whether the determinant is non-zero. Thus, being familiar with matrix algebra empowers learners to solve complex mathematical problems effectively.