Problem 54 - Precalculus with Limits

Question

In Exercises 39-54, find the determinant of the matrix.Expand by cofactors on the row or column that appears to make the computations easiest. \(\left[ \begin{array}{r} 5 & 2 & 0 & 0 & -2 \\ 0 & 1 & 4 & 3 & 2 \\ 0 & 0 & 2 & 6 & 3 \\ 0 & 0 & 3 & 4 & 1 \\ 0 & 0 & 0 & 0 & 2 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the matrix is 80.

Step 1: Identify the Matrix Type

Recognize that the given matrix is a lower triangular matrix. A lower triangular matrix has all the entries above the diagonal equal to zero.

Step 2: Calculate the determinant

Owing to the property of lower triangular matrices, where the determinant is the product of the elements on the main diagonal, calculate the determinant using the formula: \( det(A) = a_{11} * a_{22} * a_{33} * a_{44} * a_{55}\), where \(a_{ij}\) represents the element in the i-th row and j-th column.

Step 3: Get the Result

Use the elements from the matrix and calculate: \( det(A) = 5 * 1 * 2 * 4 * 2 \)

Key Concepts

Triangular Matrix

A triangular matrix is a special type of square matrix where all elements above or below the main diagonal are zero. There are two types of triangular matrices: upper triangular and lower triangular. In an upper triangular matrix, the elements below the diagonal are all zero, while in a lower triangular matrix, like the one from our exercise, the entries above the diagonal are zero.

Understanding the properties of lower triangular matrices can significantly simplify the process of finding their determinants. For such matrices, the determinant is simply the product of the elements on the diagonal. This rule applies because any row operations needed for cofactor expansion beyond the diagonal elements would involve zero for non-diagonal entries. It means we can calculate the determinant with a straightforward multiplication of diagonal terms.

Let's look at the given matrix:

• The main diagonal elements are 5, 1, 2, 4, and 2.
• The determinant is calculated as the product of these numbers.
• This considerably reduces the calculation effort required compared to non-triangular matrices.

Recognizing triangular matrices and applying their properties effectively simplifies problems and is a helpful skill in linear algebra.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, provides a method to compute the determinant of a matrix through minors and cofactors. This method is more comprehensive than directly using properties of special matrices, such as triangular or diagonal matrices, and is typically used when the matrix doesn’t fit these nice categories.

Here's a step-by-step approach to cofactor expansion:

• Select a row or column from the matrix. Ideally, choose one rich in zeros to simplify calculations.
• For each element in the chosen row or column, compute its minor. The minor is the determinant of the sub-matrix obtained by deleting the row and column of that element.
• The cofactor of an element is calculated by multiplying its minor by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices, respectively.
• Add or subtract the cofactors based on their position's sign to get the determinant.

Though not used for the lower triangular matrix in the exercise (since its properties suffice), cofactor expansion is a versatile tool.
It’s especially useful for non-triangular matrices.

Diagonal Matrix

A diagonal matrix is a simpler form of triangular matrices where all the elements outside the main diagonal are zero, not just those above or below it.
In practical terms, this means that both upper and lower non-diagonal entries are zero.

• In a diagonal matrix, computing the determinant is straightforward. It is the product of the diagonal elements, similar to triangular matrices.
• This simplicity makes diagonal matrices highly efficient for mathematical operations and computational tasks.
• Many algorithms and processes in linear algebra utilize diagonal matrices, taking advantage of their structural simplicity.

While the given exercise didn’t involve a diagonal matrix, understanding this type of matrix helps appreciate the power of simplifying matrix-related computations.
Transforming a matrix into an equivalent diagonal form can often reveal underlying characteristics and relationships vital for solving systems of equations or understanding transformations.