Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{llllr} 5 & 2 & 0 & 0 & -2 \\ 0 & 1 & 4 & 3 & 2 \\ 0 & 0 & 2 & 6 & 3 \\ 0 & 0 & 3 & 4 & 1 \\ 0 & 0 & 0 & 0 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 56.

Step 1: Identify the first row for cofactor expansion

From the given matrix, the first row is identified to be \[5, 2, 0, 0, -2\]. This row is chosen because it contains two zeros which will effectively simplify the calculations.

Step 2: Compute the determinant by cofactor expansion

Using the formula for determinant of a matrix by cofactor expansion along the first row, we have: \[\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} + a_{14}C_{14} + a_{15}C_{15}\], where \(a_{ij}\) are the elements of the first row, \(C_{ij}\) are their respective cofactors. Substituting the respective values, we obtain: \[5(-1)^{1+1}det(A_{{1,1}}) + 2(-1)^{1+2}det(A_{{1,2}})\], where \(A_{ij}\) are sub-matrices obtained by deleting the i-th row and j-th column from the original matrix. Due to zeros on third, fourth and fifth column, they won't contribute to the determinant, thus can be omitted from the computation. Hence, the determinant becomes \[5det(A_{1,1}) - 2det(A_{1,2})\].

Step 3: Calculate the determinants of the sub-matrices

We then proceed to compute the determinants of sub-matrices \(A_{1,1}\) and \(A_{1,2}\). Observe that \(A_{1,1}\) and \(A_{1,2}\) are upper triangular matrices, hence their determinant can simply be obtained by calculating the product of their diagonal elements. Hence we get det(\(A_{1,1}\)) = 1×2×4×2 = 16 and det(\(A_{1,2}\)) = 4×6×1 = 24.

Step 4: Substitute the values back to the determinant formula

Substitute the determined values back to the determinant formula: det(A) = \[5×16 - 2×24\] . Performing the multiplication and subtraction yields the result as 56.

Key Concepts

Cofactor Expansion

When calculating the determinant of a matrix, cofactor expansion is a helpful method, especially for larger matrices. This technique involves expanding the determinant across a row or column of the matrix. It's wise to choose a row or column with zeros to simplify the calculation by eliminating unnecessary terms.
The determinant, \( \text{det}(A) \), is calculated as a sum of products of the matrix elements and their cofactors. The formula for cofactor expansion along the first row is: \[ \text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + \ldots + a_{1n}C_{1n}, \] where \ (a_{ij}) \ is the element of the matrix and \ (C_{ij}) \ is the cofactor of that element. The cofactor is calculated by \ (-1)^{i+j} \ times the determinant of the sub-matrix formed by deleting the ith row and jth column. This process effectively breaks the original problem into smaller, more manageable calculations.

Upper Triangular Matrix

An upper triangular matrix is a type of matrix where all the entries below the main diagonal are zero. This special configuration simplifies the computation of a determinant.
The primary advantage of an upper triangular matrix is that its determinant is simply the product of its diagonal elements. For example, if you have an upper triangular matrix called \ (B) \, the determinant \ (\text{det}(B)) \ can be calculated as: \[ \text{det}(B) = b_{11} \times b_{22} \times \ldots \times b_{nn}, \] where \ (b_{ii}) \ are the diagonal elements of the matrix.
This principle was used in the step-by-step solution to compute the sub-matrix determinants, thereby making it much quicker to find the overall determinant of the original matrix.

Graphing Utility

Graphing utilities are handy tools often used to verify calculations such as matrix determinants, especially for students and instructors. They provide a visual way to check work and ensure accuracy.
These utilities, whether software programs or calculators with matrix functions, can input a matrix and find its determinant quickly, serving as a confirmation of manual calculations.

• Users input matrices as they appear in exercises.
• The tool helps visualize steps like expansion by cofactor.

Using a graphing utility can instill confidence in the calculated determinant and offers an opportunity to explore variations in matrix problems effortlessly.

Sub-Matrix

A sub-matrix is derived from a larger matrix by removing certain rows or columns. In the context of cofactor expansion, sub-matrices play a crucial role as they become the focus for determinant computations.
When a specific element of a matrix is selected for cofactor expansion, its cofactor involves deleting the corresponding row and column to form what is known as a sub-matrix. The determinant of this smaller matrix forms a part of the calculation in finding the larger matrix's determinant.
This approach helps break down complex determinations into smaller, more straightforward subsets, making it a valuable asset in matrix calculations in linear algebra.

• Helps manage complex matrices by reducing their size.
• Essential for calculating determinants using cofactor expansion.

Understanding sub-matrices helps in grasping how determinants reflect the larger structure of the original matrix.