Problem 16 - College Algebra

Question

For the following exercises, find the determinant. \(\left|\begin{array}{rrr}-1 & 4 & 0 \\ 0 & 2 & 3 \\ 0 & 0 & -3\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is 6.

Step 1: Identify the Type of Matrix

First, recognize that the matrix given is an upper triangular matrix. An upper triangular matrix has all elements below the main diagonal equal to zero.

Step 2: Recall Determinant Property for Upper Triangular Matrices

For an upper triangular matrix, the determinant is the product of the elements on the main diagonal. The main diagonal elements in this matrix are -1, 2, and -3.

Step 3: Calculate the Determinant

Multiply the elements on the main diagonal: \[ ext{determinant} = (-1) imes 2 imes (-3) \] Calculate this product: \[ (-1) imes 2 imes (-3) = 6 \]

Step 4: Verify the Calculation

Double-check the multiplication to ensure accuracy. The calculation: 1. Multiply (-1) by 2 to get -2. 2. Multiply -2 by (-3) to get 6. The determinant value is consistent.

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a special type of square matrix where all the elements below the main diagonal are zero. This means, for an element positioned at row \(i\) and column \(j\) in a matrix, if \(i > j\), then the element is zero.
This structuring significantly simplifies the process of finding the determinant.
Instead of going through complex calculations, you can directly use the elements on the main diagonal.
This property allows for quick determinants and is a useful shortcut in matrix math. Recognizing this matrix type can save time and avoid mistakes.

• Quick determinant calculation.
• Focus on main diagonal elements only.
• Simplifies problem-solving process.

When you identify an upper triangular matrix, you unlock this easier path to finding the determinant, which is a great advantage in mathematical operations.

Main Diagonal

The main diagonal of a matrix is an important feature in determining matrix properties like its determinant when dealing with special matrix types like the upper triangular matrix.
It consists of the elements that stretch from the top left corner to the bottom right corner of a square matrix.
In mathematical notation, the main diagonals of an \(n \times n\) matrix \(A\) are the elements \(a_{11}, a_{22}, \, \ldots, \, a_{nn}\).
For our matrix:
- The elements of the main diagonal are \(-1, 2,\) and \(-3\).
These diagonals are not merely position markers but hold crucial value in determinant calculations, especially for upper triangular and diagonal matrices.

• The diagonal connects directly to determinant calculations.
• Focuses computations on fewer components.
• Highlights the core structure of matrix.

Recognizing the main diagonal helps to quickly assess and calculate matrices' properties, especially in more simplified forms.

Matrix Multiplication

In the context of finding a determinant for an upper triangular matrix, matrix multiplication might not be needed as prominently as in other operations, but understanding this concept is still essential.
Typically, matrix multiplication involves multiplying rows by columns, which can be complex.
However, when we focus only on the main diagonal of an upper triangular matrix to find its determinant, we actually perform a simplified type of multiplication.
In our case, for the determinant:

• Multiply the first diagonal entry \(-1\) by the second entry \(2\).
• Result: \(-2\).
• Multiply \(-2\) by the final diagonal \(-3\).
• Result: \(6\).

This demonstrates how streamlined the process becomes with an upper triangular matrix.
While traditional matrix multiplication remains a broader, complex topic, recognizing when simplifications apply can make problem-solving more efficient and enable you to focus your energy where it counts most.