Problem 78 - Precalculus

Question

Determine whether each statement is true or false. $$\text { Calculate the determinant }\left|\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ 0 & b_{2} & c_{2} \\ 0 & 0 & c_{3} \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is true if given as \(a_1 \times b_2 \times c_3\).

Step 1: Understanding the Determinant of a Triangular Matrix

First, recognize that the given matrix is a triangular matrix (specifically, an upper triangular matrix) since all the elements below the main diagonal are zero. For triangular matrices, the determinant is simply the product of the diagonal elements.

Step 2: Identify Diagonal Elements

In the given matrix: \[\begin{bmatrix} a_{1} & b_{1} & c_{1} \0 & b_{2} & c_{2} \0 & 0 & c_{3} \end{bmatrix}\] the diagonal elements are \(a_1\), \(b_2\), and \(c_3\).

Step 3: Calculate the Determinant

Using the rule for the determinant of a triangular matrix, multiply the diagonal elements together: \[\text{Determinant} = a_1 imes b_2 imes c_3\].

Step 4: Conclude if Statement is True or False

Based on the calculations, the determinant of the matrix is \(a_1 imes b_2 imes c_3\). To determine if the given statement is true or false, check if the statement aligns with this result.

Key Concepts

Triangular Matrix

A triangular matrix is a special type of square matrix that simplifies many matrix calculations, particularly the determinant. There are two types of triangular matrices: upper triangular and lower triangular.
In an **upper triangular matrix**, all the entries below the main diagonal are zero. Conversely, in a **lower triangular matrix**, all entries above the main diagonal are zero.

• Upper Triangular: Non-zero entries can only be on or above the diagonal.
• Lower Triangular: Non-zero entries can only be on or below the diagonal.

This configuration makes calculations, such as finding determinants or solving equations, much easier. For instance, the determinant of a triangular matrix is straightforward to calculate. By just multiplying the diagonal elements, you can quickly determine the determinant. This property alone significantly simplifies many calculations in matrix algebra.

Matrix Algebra

Matrix algebra involves various operations that you can perform on matrices, such as addition, subtraction, multiplication, and finding determinants.
Understanding these operations is crucial in fields like engineering, computer science, and data science as they enable you to understand systems of equations and transformations.

• Addition and Subtraction: Combine matrices element-wise, meaning you add or subtract corresponding elements from each matrix.
• Multiplication: More complex than addition and involves summing the products of corresponding row and column elements.
• Determinants: A special value computed from a matrix that provides insights into many properties, such as matrix invertibility and systems of linear equations.

In the specific context of triangular matrices, matrix algebra simplifies because of the zero elements either above or below the diagonal. Calculations like finding the determinant require less effort, making triangular matrices very valuable in simplifying complex algebraic problems.

Diagonal Elements

Diagonal elements of a matrix are those that lie on the line stretching from the top-left to the bottom-right of the matrix. This diagonal plays a central role in various matrix operations, especially in calculations involving triangular matrices.
For example, in the matrix given in the problem:

• The diagonal elements are \(a_1\), \(b_2\), and \(c_3\).

In triangular matrices, these elements are particularly important because the determinant of such a matrix is simply the product of these diagonal elements. This feature makes it faster to work with these matrices in matrix algebra.
Recognizing and correctly identifying diagonal elements is crucial.

• This helps in quickly determining the determinant without additional calculations.
• Also, understanding diagonal elements supports insights into matrix properties, such as invertibility and eigenvalues.

Thus, focusing on diagonal elements is an essential part of mastering both basic and advanced concepts in linear algebra.