Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The determinant of the matrix is 80.

Step 1: Apply cofactor expansion

First, apply the cofactor expansion on the first column. As all but the first element of the first column are '0', the determinant will be quite easy to calculate. The determinant is computed as the first element in the column times its cofactor, plus the remaining elements times their respective cofactors. As the remaining elements are '0' in the first column, the determinant simplifies to 5 times its cofactor

Step 2: Determine the cofactor

The cofactor of the first element (the number '5') is the determinant of the 4x4 matrix obtained by removing the first row and column of the original 5x5 matrix. The simplified determinant is hence: \(5 \left|\begin{array}{llll} 1 & 4 & 3 & 2 \ 0 & 2 & 6 & 3 \ 0 & 3 & 4 & 1 \ 0 & 0 & 0 & 2 \end{array}\right|\) . Now, notice that the first column is again zero-filled with the exception of the first row. So applying the cofactor expansion once more, this can be simplified to \(5*1*\left|\begin{array}{lll} 2 & 6 & 3 \ 3 & 4 & 1 \ 0 & 0 & 2 \end{array}\right|\)

Step 3: Continue with Cofactor Expansion

Repeating the process, the determinant is \(5*1*2*\left|\begin{array}{ll} 4 & 1 \ 0 & 2 \end{array}\right|\). From here, it is relatively straightforward to calculate the determinant of this 2x2 matrix which is 8

Step 4: Multiply All Coefficients

Finally, multiply all the coefficients together, the determinant is given by \(5*1*2*8 = 80\)

Key Concepts

Cofactor Expansion

Cofactor expansion is a technique that simplifies the calculation of the determinant of a matrix. This method is particularly handy when a matrix has a row or column with a lot of zeros, as it reduces the computational workload significantly.

In this particular problem, we use the cofactor expansion starting with the first column of the matrix, which contains mostly zeros except for the first element. This choice is strategic because the presence of zeros minimizes the number of terms in the expansion, making calculations easier.

The process involves two main steps:

• Select a row or column to expand the determinant. Choosing one with the most zeros is optimal.
• Calculate each term by taking the element and multiplying it by its cofactor, which is itself determined by omitting its row and column to form a smaller matrix.

Remember that cofactor expansion can be applied recursively, as demonstrated in this problem, where smaller and smaller matrices are formed until a simple 2x2 matrix is reached.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with arrays of numbers and the operations on these arrays like addition, multiplication, and finding determinants. The determinant is a key aspect of matrix algebra as it provides important information about the matrix.

For example, a non-zero determinant indicates that the matrix is invertible, while a determinant of zero suggests that the matrix is singular and does not have an inverse. Understanding these properties is crucial for solving linear equations and analyzing vector spaces.

In our matrix example, matrix algebra is used by choosing strategic operations like cofactor expansion to simplify the calculations. With a 5x5 matrix, complex arithmetic operations become straightforward if we're familiar with matrix operations and properties. Recognizing the patterns and structure within matrices helps in resolving larger matrices by breaking them down recursively into simpler forms.

Properties of Determinants

Properties of determinants are foundational principles that greatly aid in solving complex matrix problems. These properties offer shortcuts and strategies, allowing us to understand matrices more intuitively.

Some key properties include:

• If a matrix has a row or a column full of zeros, its determinant is zero.
• The determinant is multiplied by any scalar value applied to a row or column, thus altering the determinant proportionally by that value.
• Switching two rows or columns changes the sign of the determinant.

These properties are important, especially in cofactor expansion, as they make certain operations either unnecessary or highlight efficient calculation methods. Identifying these properties in our exercise allowed a simplification of otherwise tedious computations, showcasing the elegance of determinants in matrix algebra.