Problem 68 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the determinant of the matrix. Do not use a graphing utility. $$ \left[\begin{array}{rrrr} 5 & 3 & 6 & 1 \\ 0 & -10 & 4 & 3 \\ 0 & 0 & 5 & 2 \\ 0 & 0 & 0 & 8 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is \(-2000\).

Step 1: Identify the type of the Matrix

Identify the type of the matrix in the problem. The given matrix is a lower triangular matrix where all entries above the main diagonal are zeros.

Step 2: Compute the Determinant

The determinant of a lower (or upper) triangular matrix is simply the product of its diagonal entries. Therefore, calculate the determinant by multiplying the diagonal elements of the matrix.

Step 3: Determinant Calculation

Calculate the determinant as \(5 * -10 * 5 * 8\).

Key Concepts

Lower Triangular Matrix

A lower triangular matrix is a unique type of square matrix. In this matrix, all elements above the main diagonal are zeros. This means every non-zero element of the matrix is either on or below the diagonal. This special characteristic makes it easier to perform certain computations, such as finding the determinant. For example, in the given matrix:

• The first row has elements both above and on the diagonal, so it is not restricted to zero, only for subsequent rows.
• From the second row onwards, all elements to the right of the diagonal are zeros.

Lower triangular matrices are commonly found in linear algebra and have a significant place in mathematical computations. Recognizing this type of matrix quickly can simplify many matrix operations.

Matrix Diagonal

The diagonal of a matrix is the set of entries extending from the top left corner to the bottom right corner. In any square matrix, these are the positions where the row index is equal to the column index, represented by elements like \(a_{11}, a_{22}, a_{33},\) etc.

For a matrix of size \(n \times n\), you will find \(n\) diagonal entries. This diagonal is important because of its properties:

• In triangular matrices, understanding the diagonal elements helps in quick calculations of certain properties, like determinants.
• They also provide insight into the eigenvalues of a matrix when it is in a special form.

In the presented matrix, the diagonal elements are 5, -10, 5, and 8.Recognizing these diagonal elements is crucial in further calculations.

Product of Diagonal Elements

The product of diagonal elements in triangular matrices, such as lower and upper triangular matrices, is directly related to the determinant. This is a simplification involved in computing the determinant thanks to the matrix structure. Instead of expanding the determinant through a complex set of operations, you can simply multiply the diagonal elements.

This principle holds because in these matrices, every row except the first influences only one factor in the determinant, simplifying the multi-step calculation to a mere multiplication:

• By using the diagonal approach, the determinant for the matrix is calculated as the product: \(5 \times (-10) \times 5 \times 8\).
• This yields the determinant as \(-2000\).

By understanding the product of diagonal elements, finding determinants for matrices like lower triangular ones becomes faster and more efficient.