Question

In Exercises \(71-74\), find the determinant of the matrix. Tell which method you used. $$ \left[\begin{array}{rrr} 3 & 0 & 0 \\ 4 & -2 & 0 \\ 5 & 4 & 3 \end{array}\right] $$

Short answer

The determinant of the matrix is -18.

Step-by-step solution

Identify the Matrix Type

The financial portfolio matrix is a lower triangular matrix. This is because all the entries above the main diagonal are zero.

Apply the Property of Determinants for Triangular Matrices

The determinant of a triangular matrix (whether upper or lower) is just the product of the entries on the main diagonal. Applying this property, the determinant of the matrix is: 3 * -2 * 3.

Calculate the determinant

The determinant is the result of multiplying the main diagonal entries: 3 * -2 * 3 = -18

Key concepts

Triangular Matrix

In the realm of linear algebra, a triangular matrix holds a special place due to its unique structure and properties. A matrix is considered triangular if all the entries either above or below its main diagonal are zero. If the zeroes are above the diagonal, it's called a lower triangular matrix; if below, an upper triangular matrix.

Why is this important? Because computations involving triangular matrices are often simpler. For instance, when solving systems of linear equations, triangular matrices make it more straightforward to use back substitution techniques. Additionally, matrix operations like finding determinants or eigenvalues become much easier, as will be further exemplified in the context of determinants.

Property of Determinants

Determinants play a crucial role in understanding the properties of a matrix. One powerful property of determinants especially relevant when dealing with triangular matrices, is that the determinant of such a matrix is simply the product of the entries on its main diagonal.

This property not only simplifies computations, but it also offers insight into the matrix's behavior. For example, it immediately tells us that if any diagonal entry of a triangular matrix is zero, the determinant is zero, which in turn indicates that the matrix is singular (not invertible). Recognizing the type of matrix you're working with can save you a considerable amount of time and effort in calculations.

Matrix Diagonal

The matrix diagonal, often referred to as the main diagonal, is the line of entries that extends from the top left corner to the bottom right corner of a square matrix. In a broader sense, the term can also refer to any of the diagonals in a rectangular matrix.

Mathematically, the main diagonal is significant because it contains elements whose row and column indices are equal, denoted as \( a_{ii} \). In many calculations, including determinants and traces of a matrix, only these diagonal elements are used. As seen in triangular matrices, the values on the main diagonal can give us direct answers to determinant-related problems without performing the full suite of determinant operations.