Problem 28 - Algebra for College Students

Question

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}1 & 2 & -3 \\ -3 & -1 & 1 \\ 4 & 5 & 4\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is 56.

Step 1: Identify the Determinant Formula

The determinant for a 3x3 matrix \( A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \) is given by: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Step 2: Assigning Values from Matrix to Formula

Assign the values from the matrix to the general equation: - \(a = 1, b = 2, c = -3\) - \(d = -3, e = -1, f = 1\) - \(g = 4, h = 5, i = 4\)

Step 3: Calculate Each Term in the Determinant Formula

Substitute the values into the formula:1. Calculate \( ei - fh \): \( (-1)(4) - (1)(5) = -4 - 5 = -9 \) 2. Calculate \( di - fg \): \( (-3)(4) - (1)(4) = -12 - 4 = -16 \) 3. Calculate \( dh - eg \): \( (-3)(5) - (-1)(4) = -15 + 4 = -11 \)

Step 4: Substitute Back to the Determinant Formula and Compute

Using the determinant formula,\[ \text{det}(A) = 1(-9) - 2(-16) + (-3)(-11) \]Compute each part:1. \( 1(-9) = -9 \) 2. \( -2(-16) = 32 \) 3. \( -3(-11) = 33 \)Combine them:\[ \text{det}(A) = -9 + 32 + 33 = 56 \]

Step 5: Verify the Calculation

Re-review the individual calculations and summation to ensure each step follows the basic arithmetic rules and calculations are accurately completed. Verified that the determinant value is 56.

Key Concepts

Determinant Calculation

Calculating the determinant of a three by three matrix can be straightforward when you understand the formula. The determinant is a special number that can provide useful information about a matrix, such as whether it has an inverse. For a 3x3 matrix, the formula is given by:

• \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \)

Here, you can see that each part involves multiplication between elements of the matrix and addition or subtraction of these products. To solve the determinant, break the matrix into smaller parts as shown:

• Calculate \( ei - fh \)
• Calculate \( di - fg \)
• Calculate \( dh - eg \)

After calculating these products, substitute back into the formula to calculate the determinant. This approach simplifies complex matrix operations into easy-to-digest steps, ensuring accuracy and clarity.

Matrix Algebra

Matrix algebra is essential for many areas of mathematics and engineering. It deals with operations on matrices which are grid-like arrays of numbers. Here's how matrix algebra is structured:

• Matrices can be added or subtracted if they're the same size.
• They can be multiplied following specific rules where the number of columns in the first matrix must equal the number of rows in the second.
• Determinants, as covered, are critical numbers derived from square matrices.

Understanding these rules allows you to manipulate matrices efficiently. The operations follow a consistent structure, making them predictable once you're familiar with the underlying concepts. This predictability is crucial when dealing with larger problems, allowing you to break down complex equations into manageable pieces.

Properties of Determinants

The properties of determinants make them a powerful tool in matrix algebra. Here are some key properties to consider:

• If any row or column of a matrix is zero, the determinant is zero.
• Swapping two rows or columns changes the sign of the determinant (i.e., the determinant becomes negative of what it was).
• Multiplying a row or column by a scalar multiplies the determinant by that scalar.
• The determinant of a matrix is zero if the matrix is singular, meaning it doesn't have an inverse.

These properties are incredibly useful in complex calculations. They allow for simplifications and, in many cases, can help you identify relationships between matrices quickly. Recognizing when a determinant simplifies can save time and effort in longer computations.