Problem 29 - Algebra and Trigonometry

Question

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{ccccc}{1} & {2} & {3} & {4} & {5} \\ {0} & {2} & {4} & {6} & {8} \\ {0} & {0} & {3} & {6} & {9} \\ {0} & {0} & {0} & {4} & {8} \\\ {0} & {0} & {0} & {0} & {5}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 120.

Step 1: Identify Upper Triangular Matrix

The given matrix is already an upper triangular matrix, meaning all elements below the main diagonal are zero. The determinant of an upper triangular matrix is the product of the diagonal elements.

Step 2: Calculate the Determinant

To find the determinant of the upper triangular matrix, multiply the diagonal elements: \[1 \times 2 \times 3 \times 4 \times 5\] Compute the product step-by-step: 1. Calculate \(1 \times 2 = 2\)2. Then \(2 \times 3 = 6\)3. Now \(6 \times 4 = 24\)4. Finally, \(24 \times 5 = 120\)

Step 3: Final Detemrinant Value

The determinant of the given matrix is the product of its diagonal elements, which is 120. Since no additional row or column operations are necessary due to the matrix's form, this is the complete determinant.

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a special type of matrix in which all the elements below the main diagonal are zero.
This kind of matrix takes a form where you can immediately notice that you only need to consider the values on the diagonal to perform certain operations, like calculating the determinant.

Imagine you have a staircase, and all the steps on the left side are missing except for those on the very path you walk. That's somewhat like our upper triangular matrix, well-organized and efficient.
Some key points about upper triangular matrices include: - Only the elements along or above the main diagonal can be non-zero. - This structure simplifies various mathematical operations, like finding determinants.
Recognizing an upper triangular matrix can save you a lot of time as it reduces the complexity of operations.

Matrix Operations

Matrix operations are mathematical processes that you perform on matrices, similar to how you add, subtract, multiply, or divide numbers.
With matrices, operations can involve combining them or changing them to make solving problems easier.

Here are some common matrix operations: - **Addition and Subtraction**: Matrices are added or subtracted element-wise, only if they have the same dimensions. - **Multiplication**: Multiplying matrices involves combining rows of the first matrix with columns of the second. - **Determinant Calculation**: The determinant is a special number that can be calculated only for square matrices. It is vital in linear algebra for solving systems of equations and evaluating matrix invertibility.
Through these various operations, you can modify matrices to gain insights or simplify computations.

Diagonal Elements

Diagonal elements of a matrix refer to the elements that are located on the line from the top left to the bottom right of a square matrix.
In a 5x5 matrix, like the one in the exercise, the diagonal elements would be those in positions \(a_{11}, a_{22}, a_{33}, a_{44}, a_{55}\).

The importance of diagonal elements is highlighted in upper triangular matrices, where the determinant can be computed just using these values.
The diagonal's simplicity turns a potentially complex operation into a straightforward process.
This property not only simplifies calculations but also aids in understanding how transformations affect space in linear algebra.
Remember: what lies on the diagonal is the key to shortcuts when evaluating matrix properties.

Product of Diagonal

Calculating the determinant of an upper triangular matrix involves taking the product of the diagonal elements.
This is a time-saving method because you don't need to perform row or column operations to simplify the matrix further.

For example, in our exercise, to evaluate the determinant: 1. Identify the diagonal elements: \(1, 2, 3, 4, 5\). 2. Multiply them step-by-step: \(1 imes 2 = 2\), \(2 imes 3 = 6\), \(6 imes 4 = 24\), and finally \(24 imes 5 = 120\).
Just like making a sandwich, you stack the layers sequentially.
Each step builds on the previous one to reach the answer efficiently.
This method makes calculating determinants of upper triangular matrices both quick and less prone to error.