Problem 21 - Differential Equations and Linear Algebra

Question

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rr}2 & -4 \\ -1 & 0\end{array}\right]\).

Step-by-Step Solution

Answer

To evaluate the determinant of the given matrix \(A=\left[\begin{array}{rr}2 & -4 \\\ -1 & 0\end{array}\right]\), multiply the elements in the main diagonal (2 and 0) and subtract the product of the elements in the other diagonal (-4 and -1). So, \(Det(A)= (2*0) - (-4*-1) = 0 - 4 = -4\). Therefore, the determinant of matrix A is -4.

Step 1: Identify the elements in the main diagonal

In the given matrix A, the elements in the main diagonal are 2 and 0.

Step 2: Identify the elements in the other diagonal

In the given matrix A, the elements in the other diagonal are -4 and -1.

Step 3: Multiply the elements in the main diagonal

Now, we need to multiply the elements in the main diagonal. 2 * 0 = 0

Step 4: Multiply the elements in the other diagonal

Next, we need to multiply the elements in the other diagonal. -4 * (-1) = 4

Step 5: Subtract the results

Now, we need to subtract the product of the other diagonal from the product of the main diagonal to calculate the determinant. Det(A) = 0 - 4 = -4 The determinant of the given matrix A is -4.

Key Concepts

Matrix Algebra

Matrix Algebra is an essential branch of mathematics that involves operations with matrices. These operations include addition, subtraction, multiplication, and finding the determinant. This powerful tool is used in various fields like physics, engineering, computer science, and economics.

One of the critical operations in matrix algebra is calculating the determinant, especially for square matrices. The determinant offers insights into the matrix’s properties, such as invertibility.

• If the determinant is zero, the matrix is singular and not invertible.
• If the determinant is non-zero, the matrix is invertible.

Calculating the determinant of a matrix, especially simple ones like the 2x2 matrix, is fundamental in grasping more complex matrix operations.

2x2 Matrix

A 2x2 matrix is a small, simple matrix with 2 rows and 2 columns. This kind of matrix is typically represented as:
\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
where \(a, b, c, \) and \(d\) are elements of the matrix. Working with 2x2 matrices gives a clear basis for understanding larger matrices and performing operations like finding determinants.

When computing the determinant of a 2x2 matrix, the formula is: \[ det(A) = a \cdot d - b \cdot c \]
In the exercise, we applied this formula using the specific elements from matrix A. The simplicity of the 2x2 matrix makes it a perfect starting point for anyone new to linear algebra.

Linear Algebra

Linear Algebra is the field of mathematics focusing on vectors, vector spaces, and linear transformations. It is foundational to many areas of applied mathematics and engineering.

One of the primary tools in linear algebra is the matrix, which simplifies computations involving linear systems and transformations. In linear algebra, knowing how to find a determinant of a matrix helps in solving linear equations and understanding the transformation properties of matrices.

• Evaluating determinants helps in solving linear systems to determine consistency.
• Determinants provide information on the eigenvalues, which is crucial in various applications like stability analysis and system dynamics.

Mastery of determinants and matrix operations is key in applying linear algebra effectively and confidently.