Problem 15 - Algebra for College Students

Question

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{ll} -2 & 5 \\ -3 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse is \( \left[ \begin{array}{cc} 2 & -\frac{5}{3} \\ 1 & -\frac{2}{3} \end{array} \right] \).

Step 1: Identify the matrix

First, we examine the given matrix, which is a 2x2 matrix: \( A = \left[ \begin{array}{ll} -2 & 5 \ -3 & 6 \end{array} \right] \). We need to find the inverse, if it exists.

Step 2: Calculate the determinant

For a 2x2 matrix \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \), the determinant is given by \( \text{det}(A) = ad - bc \). Here, \( a = -2 \), \( b = 5 \), \( c = -3 \), and \( d = 6 \). Substitute these values into the determinant formula: \( \text{det}(A) = (-2)(6) - (5)(-3) = -12 + 15 = 3 \).

Step 3: Determine if the inverse exists

A matrix has an inverse if and only if its determinant is non-zero. Since \( \text{det}(A) = 3 \), which is not zero, the matrix does have an inverse.

Step 4: Find the inverse matrix

For a 2x2 matrix \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) with determinant \( \text{det}(A) eq 0 \), the inverse is computed as \( A^{-1} = \frac{1}{\text{det}(A)} \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right] \). For our matrix, this becomes: \[ A^{-1} = \frac{1}{3} \left[ \begin{array}{cc} 6 & -5 \ 3 & -2 \end{array} \right] = \left[ \begin{array}{cc} 2 & -\frac{5}{3} \ 1 & -\frac{2}{3} \end{array} \right] \]

Step 5: Verify the inverse

To confirm that we have calculated the inverse correctly, we multiply the original matrix by its inverse. \( A \times A^{-1} = \left[ \begin{array}{cc} -2 & 5 \ -3 & 6 \end{array} \right] \times \left[ \begin{array}{cc} 2 & -\frac{5}{3} \ 1 & -\frac{2}{3} \end{array} \right] \). After performing the multiplication, ensure that the result is the identity matrix \( I = \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \). This confirms the inverse is correct.

Key Concepts

Determinant Calculation

Calculating the determinant is an essential step in finding the inverse of a matrix. For a 2x2 matrix, the determinant provides a single number that summarizes certain properties of the matrix. It is computed with the formula:

• For a matrix \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \), the determinant is \( \text{det}(A) = ad - bc \).
• To find if a matrix has an inverse, its determinant must not be zero.

In the example with a matrix \( A = \left[ \begin{array}{ll} -2 & 5 \ -3 & 6 \end{array} \right] \), calculate the determinant as follows:

• \( a = -2, \ b = 5, \ c = -3, \ d = 6 \)
• The determinant is: \( (-2)(6) - (5)(-3) = -12 + 15 = 3 \).

Since the determinant is 3, which is not zero, the matrix has an inverse.

2x2 Matrix

A 2x2 matrix is one of the most straightforward matrices in linear algebra, consisting of two rows and two columns. Its compactness makes it a perfect candidate for beginners learning about matrix operations. Here’s what you need to know:

• A 2x2 matrix is represented as \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \).
• Each element, such as \( a \), \( b \), \( c \), and \( d \), impacts the matrix’s properties, like its determinant.
• 2x2 matrices are much easier to work with compared to larger matrices, making operations like finding determinants and inverses simpler.

Understanding how to handle such matrices builds the foundation for exploring more complex structures.

Identity Matrix

The identity matrix plays a crucial role in matrix multiplication, particularly when discussing matrix inverses. It acts similarly to the number 1 in multiplication of real numbers. Key points include:

• An identity matrix for a 2x2 matrix is \( I = \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \).
• When any matrix is multiplied by its inverse, the result should be the identity matrix.
• This property confirms that the matrices are true inverses of each other.

In the exercise, after finding what is believed to be the inverse of the matrix, verifying by checking if their product equals the identity matrix confirms the correctness.

Matrix Multiplication

Matrix multiplication involves a systematic way of combining two matrices to produce a new matrix. It's essential for verifying the inverse of matrices. Here’s how it works:

• For two matrices \( A \) and \( B \), multiply each element of a row from \( A \) by each element of a column from \( B \), summing the products.
• The product matrix should be of dimensions corresponding to the outer dimensions of the multiplied matrices (e.g., a 2x2 multiplied by a 2x2 results in another 2x2 matrix).
• In our example, multiply \( \left[ \begin{array}{cc} -2 & 5 \ -3 & 6 \end{array} \right] \) by its inverse to ensure the result is \( \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \).

Mastering matrix multiplication allows for accurate verification of inverses in linear algebra.