Problem 3 - Algebra and Trigonometry

Question

Evaluate each determinant. $$Therefore, the given determinant evaluates to -29 \left|\begin{array}{rr} {-4} & {1} \\ {5} & {6} \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given 2x2 matrix is -29.

Step 1: Identify the elements of the given matrix

The given matrix is \[ \begin{array}{cc} -4 & 1 \ 5 & 6 \end{array} \]. Here, \(a = -4, b = 1, c = 5, d = 6\).

Step 2: Plug the values into the determinant formula

The determinant of a 2x2 matrix is given by the formula \(ad - bc\). Using this, we substitute the values from the given matrix to find the determinant which becomes \[ (-4)(6) - (1)(5) \].

Step 3: Perform the calculations

Performing the operations, we have \[ -24 - 5 = -29 \].

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from a square matrix. It's often denoted by putting the matrix elements inside vertical bars, similar to absolute value symbols. The determinant has important properties and plays a crucial role in linear algebra, especially when it comes to understanding linear transformations, solving systems of linear equations, and finding inverses of matrices.

For a 2x2 matrix, the determinant can be calculated using a simple formula, but for larger matrices, the process becomes more complex, involving recursion and more intricate operations. The determinant can provide critical information about the matrix, such as whether it's invertible or singular (non-invertible), and can also determine the volume scaling factor for the geometric transformations associated with the matrix.

2x2 Matrix Determinant

The determinant of a 2x2 matrix is fairly straightforward to compute. If we consider a 2x2 matrix with elements \( a, b \) in the first row and \( c, d \) in the second row, the determinant is calculated as the product of the elements on the main diagonal (top left to bottom right), \( a \) and \( d \) minus the product of the off diagonal (top right to bottom left), \( b \) and \( c \). This can be represented as \( ad - bc \).

This calculation represents the area of the parallelogram formed by the column vectors of the matrix in a two-dimensional space. A zero determinant indicates that the vectors are linearly dependent, or, in geometric terms, the vectors lie on the same line and the shape collapses to a line or a point, yielding no area.

Example Calculation

Given the matrix \[ \begin{array}{cc} -4 & 1 \ 5 & 6 \end{array} \], we identify \( a = -4 \) and \( d = 6 \) as the main diagonal and \( b = 1 \) and \( c = 5 \) as the off diagonal. Applying the formula, the determinant is \( ad - bc \), which calculates to \( (-4)(6) - (1)(5) = -24 - 5 = -29 \).

Matrix Operations

Matrix operations include a set of mathematical operations that can be performed on matrices, which include but are not limited to addition, subtraction, multiplication, and finding the determinant. These operations must follow certain rules that are consistent with the dimensions of the matrices involved.

• Addition and Subtraction: Can only be performed on matrices of the same size, resulting in a new matrix where each element is the sum or difference of the corresponding elements in the original matrices.
• Multiplication: The number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix will have dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix.
• Transpose: The rows of the original matrix become the columns of the new matrix, and vice versa.
• Inverse: Only square matrices can have an inverse and a matrix is invertible if its determinant is not zero.

When evaluating the determinant, it is crucial to correctly execute these matrix operations, as mistakes can lead to an incorrect determinant, which would significantly affect the understanding of the matrix's properties.