Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the determinant of the matrix. Do not use a graphing utility. $$ \left[\begin{array}{rrrrr} -3 & 0 & 0 & 0 & 0 \\ 4 & 1 & 0 & 0 & 0 \\ 7 & -8 & 7 & 0 & 0 \\ 6 & 4 & 0 & -2 & 0 \\ 1 & 5 & 1 & -10 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 252

Step 1: Identify Type of Matrix

Identify the type of matrix. Here, we are dealing with a lower triangular matrix. In a lower triangular matrix, all entries above the main diagonal are zero, but some entries below the diagonal may be non-zero.

Step 2: Compute Determinant of Triangular Matrix

Compute the determinant of the triangular matrix. For a triangular matrix, the determinant is the product of its diagonal elements. In this case, the diagonal elements are -3, 1, 7, -2, and 6.

Step 3: Calculate Product

Calculate the product of the diagonal elements: \(-3 \times 1 \times 7 \times -2 \times 6\).

Key Concepts

Matrix Determinant

Understanding the determinant of a matrix is fundamental in linear algebra, as it is a scalar attribute that provides invaluable insights into the properties of the matrix. When we talk about the determinant, we are referring to a complex calculation derived from a square matrix that can let us know whether or not the matrix is invertible, and hence if it has a unique solution to the associated system of linear equations.

For a 2x2 matrix, the determinant can be found by subtracting the product of its diagonals. However, as matrices grow in size, so does the complexity of calculating determinants. Determinants of larger matrices typically involve subdividing the matrix into smaller parts and using recursion through a procedure known as Laplace expansion. Nonetheless, certain matrix types, like triangular matrices, offer a shortcut to find determinants, significantly simplifying the process.

Triangular Matrix

A triangular matrix is a special type of square matrix where all the entries above or below the main diagonal are zero. If the zeros are above the main diagonal, the matrix is called a lower triangular matrix; if they're below, it is an upper triangular matrix.

The beauty of triangular matrices lies in their simplicity, especially when calculating the determinant. Another important property worth noting is that operations involving triangular matrices, such as matrix multiplication or finding the inverse, are computationally less intensive compared to other matrices. This is due to the reduced number of elements that actually contribute to the result. Because of their structure, lower and upper triangular matrices play a crucial role in various matrix decomposition methods which are used to simplify more complex linear algebra calculations.

Diagonal Elements Product

The ultimate shortcut when finding the determinant of a triangular matrix is to utilize the diagonal elements product. This means that the determinant of the matrix is simply the product of all the entries on its main diagonal.

In practice, this rule is incredibly straight-forward: for a given triangular matrix, you ignore all the non-diagonal entries and multiply together those on the diagonal. Importantly, this property holds true for both lower and upper triangular matrices, making it a universally applicable and time-saving method for these types of matrices. It is also a very common occurrence in numerical methods and algorithms that convert general matrices into triangular form precisely to exploit this simplicity in the calculation of determinants.