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. $$ \left[\begin{array}{rrrrr} 3 & 2 & 4 & -1 & 5 \\ -2 & 0 & 1 & 3 & 2 \\ 1 & 0 & 0 & 4 & 0 \\ 6 & 0 & 2 & -1 & 0 \\ 3 & 0 & 5 & 1 & 0 \end{array}\right] $$

Short answer

To give the most accurate result, further calculations have to be done on the selected 4x4 matrices \(A_1\) and \(A_2\). The determinant of the original matrix will then be calculated in Step 4 based on the results from Step 3.

Step-by-step solution

Select the column for expansion

Look at the matrix and select the column with the highest number of zeroes for expansion. The second, third, and last columns all have three zeroes. To keep it simple the last column is selected.

Expand using cofactors

Expand the determinant along the selected column, and remember to alternate the sign starting with a positive. For the given matrix, the formula will be: \(5\cdot det(A_1) - 2\cdot det(A_2)\), where \(A_1\) and \(A_2\) are the 4x4 matrices obtained by removing the first and second row, and the fifth column from the original matrix.

Solve for the determinants of the 4x4 matrices

In this step, calculate the determinants of the \(A_1\) and \(A_2\) matrices. This process may need to be repeated if the determinant cannot be calculated directly depending on the elements of these matrices.

Calculate the determinant of the original matrix

Multiply the elements of the last column by the corresponding determinants of the 4x4 matrices and sum all the terms. Make sure to alternate the sign as described in Step 2.

Confirm the result using a graphing tool

Most likely, a graphing utility can calculate determinants quite easily. Just enter the matrix into the tool and ask for the determinant. Then, compare the result from the tool with the manually calculated determinant to confirm the result.

Key concepts

Cofactor Expansion

The cofactor expansion is an efficient method used to find the determinant of a matrix, particularly when the matrix has elements of zero, as it often simplifies the computation significantly. The technique involves selecting a row or column, ideally with the most zeros, and then expanding the determinant across this row or column. You do this by multiplying each element by the determinant of its respective minor, which is the smaller matrix left when the element’s row and column are removed, and finally, applying the correct sign (positive or negative) based on its position using the checkerboard pattern of signs.

For instance, if we expand along the last column in our exercise, only two terms are left because there are three zeros. The calculation becomes simpler as we only need to consider the determinants of the resulting 4×4 matrices. This method is commonly used because it significantly reduces the amount of computation required, especially for larger matrices.

Matrix Operations

Matrix operations are the various ways in which matrices can be manipulated and are essential for working with linear equations and transformations. The determinant calculation often involves several types of matrix operations. Some of these include finding the minor matrices, which requires removing a specific row and column, and transposing a matrix, which involves switching its rows and columns.

Moreover, matrix multiplication is frequently used when expanding the determinant through the cofactor expansion method, since each element must be multiplied by the determinant of its corresponding minor. It's essential to execute these operations correctly, as even a minor error can lead to a completely wrong result for the determinant.

Graphing Utility Confirmation

Graphing utilities are powerful tools that can perform matrix operations with high accuracy and speed, providing a helpful check against manual calculations. To confirm your manually calculated determinant, you simply enter the matrix into a graphing utility and request the determinant value.

Comparing the results from the graphing tool with your hand-calculated result offers a means to verify your answer. This step is highly recommended as it can catch mistakes that might have occurred during the manual calculation process. Graphing utilities come in various forms, including graphing calculators, specialized software, and web-based apps, all designed to assist in mathematical computations.

Calculating Determinants

Calculating the determinant of a matrix is a critical procedure in linear algebra that has implications in various mathematical concepts, including understanding the solvability of systems of linear equations and finding matrix inverses. Determinants give information about the properties of the matrix, such as whether it is invertible or singular.

The process of calculating determinants can be direct for 2x2 matrices, but for larger ones, like the 5x5 in our exercise, involving recursive cofactor expansion is often necessary. This involves decomposing the matrix into smaller matrices and finding their determinants, a process that may be repeated multiple times until the determinant can be easily computed. Despite its complexity for larger matrices, understanding and executing determinant calculations is pivotal for anyone studying linear algebra or related fields.