Problem 12 - Algebra and Trigonometry

Question

Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{rr}{2.2} & {-1.4} \\ {0.5} & {1.0}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is 2.9.

Step 1: Identify the Structure of the Matrix

The input matrix is a 2x2 matrix given by \(\begin{bmatrix} 2.2 & -1.4 \ 0.5 & 1.0 \end{bmatrix} \). For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \(ad - bc\). Identify \(a = 2.2\), \(b = -1.4\), \(c = 0.5\), and \(d = 1.0\).

Step 2: Calculate the Product of the Diagonal Elements

Multiply the diagonal elements \(a\) and \(d\) of the matrix: \(2.2 \times 1.0 = 2.2\).

Step 3: Calculate the Product of the Off-Diagonal Elements

Multiply the off-diagonal elements \(b\) and \(c\): \(-1.4 \times 0.5 = -0.7\).

Step 4: Apply the Determinant Formula for a 2x2 Matrix

Substitute the results from Step 2 and Step 3 into the determinant formula: Calculate \(2.2 - (-0.7)\). This results in \(2.2 + 0.7 = 2.9\).

Step 5: Conclusion About the Determinant

The determinant of the matrix is \(2.9\), which is a non-zero value, indicating that the matrix is invertible.

Key Concepts

Matrix Operations

The process of matrix operations includes a series of calculations and manipulations that are applied to matrices. In the context of a 2x2 matrix, these operations are relatively straightforward but crucial for understanding larger matrices. Matrices are arrays of numbers arranged in rows and columns that can represent systems of linear equations, transformations, and more in mathematics and engineering.

For a 2x2 matrix, understanding the notation and structure is the first step. These matrices typically appear as follows: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix}\] where each of the components (\(a, b, c, \) and \(d\)) play specific roles in calculations, such as finding determinants. Mastery in matrix operations is required for subsequent steps, like determining if a matrix is invertible.

Calculation Steps

When faced with a 2x2 matrix, calculating the determinant is typically a straightforward task. Here’s a step-by-step breakdown:

To find the determinant of our matrix \(\begin{bmatrix} 2.2 & -1.4 \ 0.5 & 1.0 \end{bmatrix} \), follow these guidelines:

• Identify Elements: Recognize the elements where \(a = 2.2, b = -1.4, c = 0.5, \) and \(d = 1.0\).
• Multiply Diagonal Elements: Calculate \(a imes d = 2.2 imes 1.0 = 2.2\).
• Multiply Off-Diagonal Elements: Calculate \(b imes c = -1.4 imes 0.5 = -0.7\).
• Apply Determinant Formula: The determinant is given by \(ad - bc\). So perform \(2.2 - (-0.7) = 2.2 + 0.7 = 2.9\).

This determinant calculation confirms the non-zero nature, which opens the door to additional assumptions about the matrix, primarily regarding its invertibility.

Invertible Matrix

A matrix is said to be invertible if it has an inverse; this is a matrix which, when multiplied with the original, results in the identity matrix. Passing the determinant test is a key factor in determining inversion capability: a matrix with a non-zero determinant can be inverted.

In our scenario, once we found the determinant of the given matrix to be \(2.9\), it unequivocally demonstrated the matrix is invertible. This particular characteristic is crucial in many applications, such as solving systems of linear equations and performing matrix-based transformations.

In simpler terms, an invertible matrix allows for the reversal of transformations applied to data. This can be visualized in applications like graphic simulations where objects are rotated, scaled, or translated through matrices, and one might need to revert these operations to their original states. Always verify if a matrix is invertible before attempting inversion procedures, as missteps can lead to mathematical errors or undefined results.