Question

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

Short answer

The determinant of the matrix is -6

Step-by-step solution

Identify the type of the matrix

The matrix provided is an upper triangular matrix. This type of matrix has non-zero entries along the main diagonal, starting from the upper left-hand corner. The rest of the entries below this diagonal are zero.

Find the determinant of the matrix

For any triangular matrix (upper or lower), the determinant is the product of the entries on the main diagonal. Therefore, the determinant is given by $det(Matrix)=2\ast (-3)\ast 1$

Evaluate the product

Evaluate the product that was derived in the previous step: $det(Matrix)=-6$

Key concepts

Upper Triangular Matrix

When learning about matrices in algebra, an upper triangular matrix stands out because of its particular structure. This type of matrix is square, having the same number of rows and columns, and is characterized by having all non-zero elements on or above the main diagonal, while all elements below this diagonal are zero. Formally, for any entry in an upper triangular matrix denoted as $a_{ij}$, where $i$ is the row index and $j$ is the column index, $a_{ij}=0$ for all $i>j$.

The advantage of working with upper triangular matrices becomes evident when performing certain matrix operations, such as calculating determinants. The structure allows for simpler computations, saving time and reducing the potential for errors. In practical applications, solving systems of linear equations through methods like back substitution leverages the benefits of upper triangular matrices. Understanding how to work with these matrices is fundamental for further studies in linear algebra and its applications in fields such as engineering and computer science.

Main Diagonal

The main diagonal of a matrix plays a pivotal role in matrix algebra. It is the diagonal line of entries that runs from the upper left to the lower right corner of a square matrix. In a matrix, the entries on the main diagonal are denoted by $a_{ii}$, where the row index and column index are identical. For example, in a 3x3 matrix, the main diagonal would consist of the elements $a_{11},a_{22},$ and $a_{33}$.

In an upper triangular matrix, these main diagonal entries are of significant importance because they hold the key to calculating the determinant. This makes it crucial to understand the concept of the main diagonal and be able to identify it easily. In subjects such as physics and engineering, the main diagonal can represent the principal components of a system, highlighting their foundational significance.

Evaluate Determinant

Assessing the determinant of a matrix, which we denote as $det(A)$ for a given matrix $A$, is a fundamental operation in linear algebra. It is a single number that provides valuable information about the matrix, including whether a set of linear equations has a unique solution and whether the matrix is invertible.

For an upper triangular matrix, the process to evaluate the determinant is notably simplified because it is equal to the product of the entries on the main diagonal. There is no need to perform row operations or expansion by minors, which are needed for more complex matrices. This ease of calculation underscores the importance of recognizing matrix types, as it directly affects the practicality of mathematical computations. The determinant's value can also influence the behavior of a system of linear equations; for example, if the determinant is zero, the matrix will not have an inverse, and the system of equations it represents may not have a unique solution. Understanding the determinant gives insight into the geometry of transformations represented by matrices, making it a fundamental concept in mathematics and related disciplines.