Question

Find the inverse of the given matrix \(\mathbf{A}\) in each of Exercises 13-24. $$ \mathbf{A}=\left(\begin{array}{rr} 4 & -2 \\ -6 & -1 \end{array}\right) $$

Short answer

The inverse of matrix \(\mathbf{A}\) is: $$ \mathbf{A}^{-1}=\left(\begin{array}{rr} \frac{1}{16} & -\frac{1}{8} \\ -\frac{3}{8} & -\frac{1}{4} \end{array}\right) $$

Step-by-step solution

Calculate the determinant

Find the determinant of matrix \(\mathbf{A}\), which is given by the formula \(\text{det} (\mathbf{A}) = (a \times d) - (b \times c)\), where \(a\), \(b\), \(c\), and \(d\) are the matrix elements in a 2x2 matrix. $$ \text{det} (\mathbf{A}) = (4 \times(-1)) - ((-2) \times (-6)) $$

Determine if the inverse exists

Calculate the determinant and check if it's nonzero. $$ \text{det} (\mathbf{A}) = -4 - 12 = -16 $$ Since the determinant is nonzero, the inverse of the matrix \(\mathbf{A}\) exists.

Swap the top-left and bottom-right elements

Swap the top-left element (4) with the bottom-right element (-1) in matrix \(\mathbf{A}\) to create a new matrix. $$ \mathbf{A'}=\left(\begin{array}{rr} -1 & -2 \\ -6 & 4 \end{array}\right) $$

Change the signs of the top-right and bottom-left elements

Change the signs of the top-right element (-2) and the bottom-left element (-6) in matrix \(\mathbf{A'}\). $$ \mathbf{A''}=\left(\begin{array}{rr} -1 & 2 \\ 6 & 4 \end{array}\right) $$

Divide each element by the determinant

Divide each element of the matrix \(\mathbf{A''}\) by the determinant, -16. $$ \mathbf{A}^{-1} = \frac{1}{\text{det}(\mathbf{A})} \mathbf{A''} = \frac{1}{-16} \left(\begin{array}{rr} -1 & 2 \\ 6 & 4 \end{array}\right) $$ Finally, we find the inverse matrix, \(\mathbf{A}^{-1}\). $$ \mathbf{A}^{-1}=\left(\begin{array}{rr} \frac{1}{16} & -\frac{1}{8} \\ -\frac{3}{8} & -\frac{1}{4} \end{array}\right) $$

Key concepts

Matrix Determinant

The determinant of a matrix is a special number that provides important information about the matrix. It is especially useful when working with a 2x2 matrix, as it helps us determine the existence of an inverse. For a 2x2 matrix \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \] the determinant is calculated using the formula: \[ \text{det}(A) = (a \times d) - (b \times c) \] This formula can be remembered by multiplying the diagonal elements and subtracting the product of the off-diagonal elements.

• In our example, matrix \(\mathbf{A}\) has the elements \(a=4\), \(b=-2\), \(c=-6\), and \(d=-1\).
• So, the determinant \( \text{det}(\mathbf{A}) = (4 \times -1) - ((-2) \times -6) \).

By calculating these products, we find that \(\text{det}(\mathbf{A}) = -4 - 12 = -16\). When a determinant is non-zero, like in this exercise, it indicates that an inverse matrix exists.

2x2 Matrix

In linear algebra, a 2x2 matrix is a simple structure that consists of four numbers arranged in two rows and two columns. This is the smallest square matrix possible and is often used in basic matrix operations.

• A 2x2 matrix is represented generally as \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \).
• This pattern means it has two rows \([a, b]\) and \([c, d]\).

When dealing with 2x2 matrices, swap the diagonal and swap the signs of the off-diagonal elements during inversion. For example, to invert matrix \( \mathbf{A} \):

• Simply swap element \(a\) with \(d\) and negate \(b\) and \(c\).
• This process creates a new intermediate matrix from which you form the inverse.

These steps can look complicated at first, but they quickly become straightforward with practice. By understanding the basic properties of a 2x2 matrix, solving related problems becomes much easier.

Matrix Algebra

Matrix algebra involves various techniques that allow us to perform operations such as addition, subtraction, multiplication, and finding inverses with matrices. It is an essential tool in linear algebra used to solve systems of equations and transform data. In this exercise, we are particularly focused on the process of finding an inverse matrix. Finding the inverse of a 2x2 matrix requires several specific steps in matrix algebra:

• First, calculate the determinant to ensure the matrix can be inverted.
• Next, adjust the current matrix by performing swaps and sign changes as outlined.
• Finally, divide each element of the modified matrix by the computed determinant.

The resultant matrix is the inverse, which, when multiplied by the original matrix, yields the identity matrix: \[\mathbf{A} \times \mathbf{A}^{-1} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \] By following these steps, matrix algebra provides us the ability to not only solve equations but also to understand transformations and other intricate operations.