Problem 16 - A Graphical Approach to College Algebra

Question

Find each determinant. $$\operatorname{det}\left[\begin{array}{ccc}2 & 1 & -1 \\\4 & 7 & -2 \\\2 & 4 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 10.

Step 1: Write the matrix and formula

We need to find the determinant of a 3x3 matrix. The matrix is given by \(A = \begin{bmatrix} 2 & 1 & -1 \ 4 & 7 & -2 \ 2 & 4 & 0 \end{bmatrix}\). The formula for the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is \( \operatorname{det}(A) = a(ei−fh) - b(di−fg) + c(dh−eg) \).

Step 2: Identify elements of the matrix

Identify each of the elements in the determinant formula: \( a = 2, b = 1, c = -1, d = 4, e = 7, f = -2, g = 2, h = 4, i = 0 \).

Step 3: Substitute elements into the formula

Substitute the elements into the determinant formula:\[\operatorname{det}(A) = 2(7 \cdot 0 - (-2) \cdot 4) - 1(4 \cdot 0 - (-2) \cdot 2) + (-1)(4 \cdot 4 - 7 \cdot 2)\].

Step 4: Simplify calculations inside parentheses

Calculate each of the products inside the determinants:- \(7 \cdot 0 = 0\) and \((-2) \cdot 4 = -8\), so \(7 \cdot 0 - (-2) \cdot 4 = 8\).- \(4 \cdot 0 = 0\) and \((-2) \cdot 2 = -4\), so \(4 \cdot 0 - (-2) \cdot 2 = 4\).- \(4 \cdot 4 = 16\) and \(7 \cdot 2 = 14\), so \(4 \cdot 4 - 7 \cdot 2 = 2\).

Step 5: Compute final determinant value

Now substitute back into the simplified expression:\[\operatorname{det}(A) = 2 \cdot 8 - 1 \cdot 4 + (-1) \cdot 2 = 16 - 4 - 2 = 10\].

Step 6: Conclusion

The determinant of the matrix \(A\) is \(10\). Hence, \(\operatorname{det}\left(\begin{bmatrix} 2 & 1 & -1 \ 4 & 7 & -2 \ 2 & 4 & 0 \end{bmatrix}\right) = 10\).

Key Concepts

Understanding a 3x3 Matrix

A 3x3 matrix is a simple type of square matrix composed of three rows and three columns. Matrices are arrays of numbers used to organize data or equations. In a 3x3 matrix, there are nine elements represented typically as letters based on their position.- Consider a generic matrix: \ \[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\] Each letter from 'a' to 'i' corresponds to different positions within the matrix.- The goal when working with matrices often involves finding the determinant.- Determining characteristics like invertibility or solvability of systems often depend on this value.In our exercise, the matrix given is:\[\begin{bmatrix}2 & 1 & -1 \4 & 7 & -2 \2 & 4 & 0\end{bmatrix}\] Each specific number represents unique data values that can be worked with through matrix operations.

Determinant Calculation Made Easy

The determinant of a matrix is a special number that can provide a lot of information about the matrix. For a 3x3 matrix, a determinant is calculated using a specific formula which involves the values present in the matrix's elements.- The basic steps to calculate the determinant of a 3x3 matrix, given: \[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i\end{bmatrix}\] - Start by identifying each element (a to i). - Use the formula: - \(\operatorname{det}(A) = a(ei−fh) - b(di−fg) + c(dh−eg)\) - Substitute each matrix element into the formula. - Simplify the calculations to find the determinant.This determinant can inform if a matrix is invertible (non-zero determinant) or singular (if determinant is zero). Here, simpler operations inside the parentheses such as multiplication and subtraction dictate steps of simplification.

A Brief Overview of Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and their operations. It is widely used in various fields including computer science, economics, and engineering. - **Key Operations:** - Addition and subtraction: Requires matrices of the same dimensions. - Scalar multiplication: Involves multiplying each matrix element by a scalar (number). - Matrix multiplication: More complex, not commutative, order matters. - **Role of Determinants:** - Helps determine if matrices have inverses. - Used in solving systems of linear equations through methods such as Cramer's Rule. Learning matrix algebra empowers one to handle problems involving data structures. It becomes easier to transform systems of equations into simpler forms for solving real-world problems. Calculating determinants is a foundational skill, aiding in uncovering these deeper aspects of matrix math.