Problem 15 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The determinant is 17.

Step 1: Understanding the Determinant Formula for a 3x3 Matrix

To find the determinant of a 3x3 matrix, we use the formula: \[\operatorname{det}\begin{bmatrix}\ a & b & c \d & e & f \g & h & i \end{bmatrix} = a(ei-fh) - b(di-fg) + c(dh-eg) \] Here, each letter represents a different entry in the matrix, starting from the top-left corner as \(a\) and ending at the bottom-right corner as \(i\).

Step 2: Identify Elements in the Matrix

Given the matrix \(\begin{bmatrix}1 & 2 & 0 \-1 & 2 & -1 \0 & 1 & 4\end{bmatrix} \), identify the elements corresponding to the formula:- \(a = 1\), \(b = 2\), \(c = 0\)- \(d = -1\), \(e = 2\), \(f = -1\)- \(g = 0\), \(h = 1\), \(i = 4\)

Step 3: Compute Each Part of the Determinant Formula

Compute each of the three parts separately using the identified elements:- For \(a(ei - fh)\): Calculate \(ei - fh = 2\times4 - (-1)\times1 = 8 + 1 = 9\), so \(a(ei - fh) = 1 \times 9 = 9\).- For \(-b(di - fg)\): Calculate \(di - fg = (-1)\times4 - 0\times1 = -4\), so \(-b(di - fg) = -2 \times (-4) = 8\).- For \(c(dh - eg)\): Since \(c = 0\), \(c(dh - eg) = 0\times (..some value..) = 0\).

Step 4: Sum the Results

Add the results of the three computations together to find the determinant:- \(9 + 8 + 0 = 17\).

Step 5: Conclude the Determinant Calculation

The determinant of the matrix is calculated as \(17\). Thus, the determinant of the given matrix is \(17\).

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices, which are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. It's an essential tool in various fields such as physics, computer science, and economics. In matrix algebra, you can perform operations like addition, subtraction, and multiplication.
However, one of the most interesting operations is finding the determinant, which can give you valuable insights about the matrix.

• Determinants can tell you if a matrix is invertible, meaning if it has an inverse matrix.
• If the determinant equals zero, the matrix is not invertible.
• The determinant can help solve systems of linear equations.

When working with matrices, learning these algebraic manipulations can greatly simplify problem-solving.

3x3 Matrix

A 3x3 matrix is a specific type of matrix with three rows and three columns. Each element in a 3x3 matrix is usually labelled with a letter, often from the alphabet, and by convention, the elements of the matrix are denoted as follows:
\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]
In a 3x3 matrix, there are a total of nine elements, each with its designated position. The first row contains \(a, b, c\), the second row has \(d, e, f\), and the third row comprises \(g, h, i\).

• Matrix representation aids in organizing information systematically.
• The arrangement helps in applying operations methodically, such as determinant calculation.
• Knowing the position of each element is crucial for applying matrix formulas.

Complex operations like computing determinants or finding eigenvalues often start by focusing on such sliced-up, orderly presentations.

Determinant Calculation

The determinant calculation of a 3x3 matrix allows you to understand important properties about the matrix, such as whether it's invertible. Calculating the determinant involves a specific formula. For a matrix:
\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]
The formula used is:
\[ \operatorname{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]This breaks down the calculation into manageable parts by focusing on smaller 2x2 matrices created from the elements.

• First, compute \(ei-fh\) using elements from the matrix.
• Then compute \(di-fg\).
• Next, compute \(dh-eg\).
• Combine these calculations with coefficients \(a, -b, c\).
• Finally, sum these up to get your determinant.

Breaking down the formula in such a way makes it easier to follow, given that each step focuses on a smaller subset of matrix elements. This formula provides a systematic approach to a seemingly complex task.