Problem 23 - College Algebra

Question

Evaluate each determinant. $$\left|\begin{array}{rrr}3 & 0 & 0 \\\2 & 1 & -5 \\\2 & 5 & -1\end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the matrix is -3.

Step 1: Identify the type of matrix

Look at the matrix and identify that it is a upper triangular matrix.

Step 2: Apply the rule for the determinant of triangular matrices

For an upper triangular matrix, the determinant is the product of the diagonal elements. Therefore, for this matrix, multiply the elements on the diagonal, which are 3, 1, and -1.

Step 3: Calculate the determinant

Multiply the elements from step 2 together: \(3\ * 1\ *-1 = -3\).

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a special type of square matrix. The key feature of an upper triangular matrix is that all elements below the main diagonal are zero. This means that in an upper triangular matrix, you won't find any number other than zero in positions beneath the diagonal. For example, in our provided matrix, the entries in the first row are all right-aligned and non-zero, the second row has a zero before its first number, and the third row has zeros before the first and second numbers. This structure makes upper triangular matrices particularly useful in simplifying many mathematical operations.

• The word 'upper' indicates that the significant numbers lie in the upper part of the matrix.
• Being square means having the same number of rows and columns.

Working with triangular matrices like these simplifies the calculation of their determinants, which we will discuss further.

Matrix Multiplication

Matrix multiplication is a process of combining two matrices to produce a third matrix. The process involves multiplying the rows of the first matrix by the columns of the second. However, in the context of finding the determinant of an upper triangular matrix, this concept becomes more straightforward. There is no need to perform typical matrix multiplication, as we focus only on multiplying specific elements of the matrix. Specifically, to find the determinant, we multiply the diagonal elements together. This doesn't mean you're performing standard matrix multiplication, but rather you're applying a specific method to compute the determinant in a simplified manner.

• For our matrix, apply this by directly multiplying the diagonal numbers.
• This approach exploits the properties of the triangular matrix for easier calculation.

This makes matrix multiplication different in this instance, restricted to a simplified product of elements.

Diagonal Elements

Diagonal elements in a matrix are key to many operations, especially when dealing with determinants. In a square matrix, the diagonal elements are those that form a sequence starting from the top left to the bottom right corner. These play a central role in upper triangular matrices. To calculate the determinant of such a matrix, you primarily pay attention to these diagonal elements. In our matrix, the diagonal elements are 3, 1, and -1.

• The first diagonal element is in the first row, first column.
• The second diagonal element is in the second row, second column.
• The third diagonal element is in the third row, third column.

By multiplying these diagonal elements together, you efficiently determine the overall determinant of the matrix. For the matrix here, it results in a product of 3, 1, and -1, giving us a determinant of -3. This method is not only fast but also reduces the complexity that usually accompanies more elaborate matrix operations.