Problem 10 - 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. $$\left|\begin{array}{llll} 2 & 1 & 3 & 5 \\ 3 & 0 & 1 & 2 \\ 4 & 1 & 4 & 3 \\ 5 & 2 & 5 & 3 \end{array}\right|$$

Step-by-Step Solution

Answer

We have used elementary row operations to convert the given matrix A into the upper triangular form Matrix B: $$\left|\begin{array}{llll} 2 & 1 & 3 & 5 \\\ 0 & 1 & \frac{5}{3} & \frac{5}{3} \\\ 0 & 0 & \frac{1}{3} & -\frac{16}{3} \\\ 0 & 0 & -2 & -8 \end{array}\right|$$ The determinant of an upper triangular matrix can be calculated by multiplying the main diagonal elements. Thus, \(\text{Det}(A) = -\frac{16}{3}\).

Step 1: Row Operations to Convert into Upper Triangular Form

Apply elementary row operations to convert matrix A into upper triangular form. 1. Add \((-\frac{3}{2} R_1)\) to row 2 (Replace R2 by \(R2 - \frac{3}{2} R1\)): $$\left|\begin{array}{llll} 2 & 1 & 3 & 5 \\\ 0 & -\frac{3}{2} & -\frac{5}{2} & -\frac{5}{2} \\\ 4 & 1 & 4 & 3 \\\ 5 & 2 & 5 & 3 \end{array}\right|$$ 2. Add \((-2 R_1)\) to row 3 (Replace R3 by \(R3 - 2 R1\)): $$\left|\begin{array}{llll} 2 & 1 & 3 & 5 \\\ 0 & -\frac{3}{2} & -\frac{5}{2} & -\frac{5}{2} \\\ 0 & -1 & -2 & -7 \\\ 5 & 2 & 5 & 3 \end{array}\right|$$ 3. Add \((-\frac{5}{2} R_1)\) to row 4 (Replace R4 by \(R4 - \frac{5}{2} R1\)): $$\left|\begin{array}{llll} 2 & 1 & 3 & 5 \\\ 0 & -\frac{3}{2} & -\frac{5}{2} & -\frac{5}{2} \\\ 0 & -1 & -2 & -7 \\\ 0 & -\frac{1}{2} & -\frac{5}{2} & -\frac{19}{2} \end{array}\right|$$ 4. Multiply row 2 by \((-\frac{2}{3})\): $$\left|\begin{array}{llll} 2 & 1 & 3 & 5 \\\ 0 & 1 & \frac{5}{3} & \frac{5}{3} \\\ 0 & -1 & -2 & -7 \\\ 0 & -\frac{1}{2} & -\frac{5}{2} & -\frac{19}{2} \end{array}\right|$$ 5. Add row 2 to row 3 (Replace R3 by \(R3 + R2\)): $$\left|\begin{array}{llll} 2 & 1 & 3 & 5 \\\ 0 & 1 & \frac{5}{3} & \frac{5}{3} \\\ 0 & 0 & \frac{1}{3} & -\frac{16}{3} \\\ 0 & -\frac{1}{2} & -\frac{5}{2} & -\frac{19}{2} \end{array}\right|$$ 6. Add \((\frac{1}{2} R_2)\) to row 4 (Replace R4 by \(R4 + \frac{1}{2} R2\)): $$\left|\begin{array}{llll} 2 & 1 & 3 & 5 \\\ 0 & 1 & \frac{5}{3} & \frac{5}{3} \\\ 0 & 0 & \frac{1}{3} & -\frac{16}{3} \\\ 0 & 0 & -2 & -8 \end{array}\right|$$ The matrix is now in upper triangular form. Call this upper triangular matrix as B.

Step 2: Calculate the Determinant

The determinant of an upper triangular matrix can be calculated by multiplying the diagonal elements (the main diagonal). Matrix B's diagonal elements are: 2, 1, \(\frac{1}{3}\), and -8. Thus, $$\text{Det}(B) = 2 \times 1 \times \frac{1}{3} \times (-8) = -\frac{16}{3}$$ Therefore, the determinant of the given matrix A is: $$\text{Det}(A) = -\frac{16}{3}$$

Key Concepts

Elementary Row Operations

Elementary row operations are crucial when it comes to transforming matrices into a desired form without altering their determinant. These operations help simplify matrix calculations, especially when you need to find determinants or solve systems of equations. Here's a quick breakdown of the three types of elementary row operations:

• Row switching: Swapping two rows within the matrix. This operation changes the sign of the determinant.
• Row multiplication: Multiplying every element of a row by a non-zero scalar. The determinant gets multiplied by the scalar as well.
• Row addition: Adding a multiple of one row to another row. This operation doesn't affect the determinant.

In our example, several elementary row operations were applied to simplify the matrix to an upper triangular form. By carefully selecting which operation to apply, you can systematically zero out elements below the main diagonal. Remember, the ultimate goal is to achieve a form where it's easier to compute the determinant.

Upper Triangular Matrix

An upper triangular matrix is a type of square matrix where all the elements below the main diagonal are zero. This pattern makes calculations, particularly determinants, much more straightforward. In an upper triangular matrix, you only focus on the diagonal elements to determine the whole matrix's determinant.

In the given problem, the original matrix was carefully manipulated using elementary row operations to become an upper triangular matrix. This was crucial because it simplifies the computation of the determinant by reducing complex multi-element operations into straightforward multiplications of the diagonal elements.

Transforming a matrix into an upper triangular form is also beneficial for solving systems of linear equations through methods such as back-substitution. The main advantage is evident in the reduction of computational complexity, thereby making otherwise intricate problems much more manageable.

Matrix Diagonal Elements

Diagonal elements in a matrix refer to those that run from the top left to the bottom right corner of a square matrix. In mathematical terms, the diagonal consists of elements a_{11}, a_{22}, ..., a_{nn}, where the row and column indices are equal.

When you have an upper triangular matrix, the calculation of its determinant becomes a matter of multiplying these diagonal elements together. Such was the case in our exercise example, where after converting the matrix to upper triangular form, only the elements 2, 1, \(\frac{1}{3}\), and -8 were necessary to find the determinant.

The simplicity of the diagonal elements greatly aids in reducing the effort required to solve many problems and highlights the elegance of linear algebra in simplifying complex issues. Always remember, the diagonal holds the key to solving for determinants efficiently when dealing with upper triangular matrices.