Problem 11 - Differential Equations and Linear Algebra

Question

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrrr} 2 & -1 & 3 & 4 \\ 7 & 1 & 2 & 3 \\ -2 & 4 & 8 & 6 \\ 6 & -6 & 18 & -24 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the given matrix is 2200 after using elementary row operations to reduce it to upper triangular form and then multiplying the diagonal elements.

Step 1: Row operations to reduce the matrix to upper triangular form

The given matrix is: $$\left|\begin{array}{rrrr} 2 & -1 & 3 & 4 \\\ 7 & 1 & 2 & 3 \\\ -2 & 4 & 8 & 6 \\\ 6 & -6 & 18 & -24 \end{array}\right|$$ We'll perform the following row operations to transform it into an upper triangular matrix: \(R_2 = R_2 - \frac{7}{2}R_1\) \(R_3 = R_3 + R_1\) \(R_4 = R_4 - 3R_1\) After performing these row operations, we get: $$\left|\begin{array}{rrrr} 2 & -1 & 3 & 4 \\\ 0 & \frac{5}{2} & -\frac{13}{2} & -\frac{11}{2} \\\ 0 & 3 & 11 & 10 \\\ 0 & -3 & 9 & -36 \end{array}\right|$$ Now, we'll perform the following row operations to further simplify the matrix: \(R_4 = R_4 + R_3\) After performing this row operation, we get: $$\left|\begin{array}{rrrr} 2 & -1 & 3 & 4 \\\ 0 & \frac{5}{2} & -\frac{13}{2} & -\frac{11}{2} \\\ 0 & 3 & 11 & 10 \\\ 0 & 0 & 20 & -26 \end{array}\right|$$ Now, we have an upper triangular matrix.

Step 2: Calculate the determinant

The determinant of an upper triangular matrix can be found by simply multiplying the diagonal elements. Let's calculate the determinant: $$\text{det} = 2 \times \frac{5}{2} \times 11 \times 20$$ After simplifying, we get: $$\text{det} = 2200$$ And so the determinant of the given matrix is 2200.

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a type of square matrix where all the elements below the main diagonal are zero. The main diagonal of a matrix consists of elements from the top left corner to the bottom right corner. Upper triangular matrices are significant in linear algebra because they make many matrix operations, particularly the calculation of determinants, simpler and more efficient.

For example, consider the matrix after we've performed certain row operations on the original matrix:
\[\left|\begin{array}{cccc}2 & -1 & 3 & 4 \0 & \frac{5}{2} & -\frac{13}{2} & -\frac{11}{2} \0 & 3 & 11 & 10 \0 & 0 & 20 & -26 \end{array}\right|\]This is now an upper triangular matrix. Notice how all entries below the diagonal are zero. Because of this structure, we can easily calculate the determinant by multiplying the diagonal elements.

Elementary Row Operations

In linear algebra, elementary row operations are simple manipulations that can be performed on the rows of a matrix. They are tools to transform matrices into easier forms without altering the intrinsic properties of the system they represent. There are three types of operations:

• Swap the position of two rows (\(R_i \leftrightarrow R_j\)).
• Multiply a row by a non-zero scalar (\(kR_i\)).
• Add a scalar multiple of one row to another (\(R_i + kR_j\)).

These operations are used for various purposes such as solving linear equations, finding inverse matrices, and simplifying matrices to calculate determinants, as in our example with the given matrix. The chosen operations transform the initial matrix into an upper triangular matrix, without changing its determinant.

Matrix Determinant

The determinant of a matrix is a special scalar value that provides important information about the matrix and the linear system it represents. It can tell us, for instance, whether a set of linear equations has a unique solution, if the matrix is invertible, and the volume of geometrical shapes in linear transformations.

In the case of upper triangular matrices, the calculation of the determinant is greatly simplified. It is equal to the product of the diagonal elements of the matrix. For the given upper triangular matrix, the determinant is calculated as follows:\[\text{det} = 2 \times \frac{5}{2} \times 11 \times 20\]The rules for finding the determinant differ if the matrix isn't in upper triangular form, and often require more complex operations or expansion along rows or columns.

Linear Algebra

As a branch of mathematics, linear algebra is fundamental to many areas of both pure and applied science. It focuses on the study of vector spaces, linear mappings, and systems of linear equations. In essence, it's the language of vectors and matrices. Linear algebra helps us understand and manipulate these structures to solve real-world problems ranging from engineering to economics.

Mastering the concepts of elementary row operations, matrix determinants, and understanding special matrices such as the upper triangular matrix, equip students with the tools to engage in more complex linear algebra topics like eigenvalues and eigenvectors, orthogonality, and spectral theory. These concepts are not just abstract notions but are applied in areas such as physics, computer graphics, statistics, and machine learning.