Problem 9 - Algebra for College Students

Question

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{ll} -3 & 2 \\ -4 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse is \( \begin{bmatrix} -\frac{5}{7} & \frac{2}{7} \\ -\frac{4}{7} & \frac{3}{7} \end{bmatrix} \).

Step 1: Identify the Matrix

The matrix we are working with is \[ \begin{bmatrix} -3 & 2 \ -4 & 5 \end{bmatrix} \]

Step 2: Check if the Matrix is Invertible

A 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is invertible if its determinant is non-zero. Compute the determinant using the formula: \( ad - bc \). For our matrix, the determinant is:\(-3 \times 5 - (-4) \times 2 = -15 + 8 = -7\). Since the determinant is not zero, the matrix is invertible.

Step 3: Find the Formula for the Inverse

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), if the determinant \( (ad-bc) eq 0 \),the inverse is given by: \[ \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \].

Step 4: Compute the Inverse

Since we have the determinant as \(-7 \), the inverse matrix is calculated as:\[ \frac{1}{-7} \begin{bmatrix} 5 & -2 \ 4 & -3 \end{bmatrix} \] which simplifies to:\[ \begin{bmatrix} -\frac{5}{7} & \frac{2}{7} \ -\frac{4}{7} & \frac{3}{7} \end{bmatrix} \].

Key Concepts

Determinant Calculation

Calculating the determinant is the first essential step to understand if a matrix is invertible. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is found using the formula \( ad - bc \). This mathematical expression involves simple multiplication and subtraction. By multiplying the diagonals (major diagonal: \( a \) and \( d \), minor diagonal: \( b \) and \( c \)), you obtain values that you subtract from each other.

The determinant provides valuable information about the matrix. A non-zero determinant means the matrix has an inverse, while a determinant of zero indicates it's non-invertible or singular. In the example provided, the determinant is \(-3 \times 5 - (-4) \times 2 = -15 + 8 = -7\). This non-zero result confirms the matrix is invertible.

2x2 Matrix

A 2x2 matrix is a square matrix that has two rows and two columns. These matrices are the simplest non-trivial square matrices and serve as a fundamental building block in linear algebra.

The general form of a 2x2 matrix is:

• First row contains: \(a\) and \(b\)
• Second row contains: \(c\) and \(d\)

These matrices are prevalent because of their simplicity and their ability to demonstrate basic matrix operations such as addition, multiplication, and finding inverses. Solving a problem involving a 2x2 matrix requires understanding of manipulation and matrix properties.

Invertibility of Matrices

The invertibility of a matrix refers to the existence of its inverse. A matrix is invertible if there exists another matrix that, when multiplied with the original, results in the identity matrix. For 2x2 matrices, the condition for invertibility is that the determinant should not be zero.

If the determinant \((ad - bc) eq 0\), you can find the inverse using a specific formula. This concept is crucial because it helps in solving linear equations and transformations. In our exercise, since the given matrix has a determinant of \(-7\), it is indeed invertible, and we can calculate its inverse as explained further.

Matrix Operations

Matrix operations involve the manipulation of matrices through various mathematical procedures. The basic operations include addition, subtraction, multiplication, and finding the inverse. These operations have specific properties and rules that must be followed.

For instance, the inverse of a 2x2 matrix is calculated with the formula: \[ \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \. \] This operation requires the determinant, as described earlier, to be non-zero.

Understanding these operations is vital as they provide the ability to solve complex linear algebra problems and comprehend transformations. These operations are applicable in various fields, including computer graphics, physics, and engineering, highlighting the importance of mastering them.