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Chapter 1: Problem 17

Use the given information to find \(A\). $$(I+2 A)^{-1}=\left[\begin{array}{rr} -1 & 2 \\ 4 & 5 \end{array}\right]$$

Short Answer

Expert verified
The matrix \(A\) is \(\begin{pmatrix} -\frac{9}{13} & \frac{1}{13} \\ \frac{2}{13} & -\frac{6}{13} \end{pmatrix}.\)

Step by step solution

01

Assessing the Inverse Matrix

The matrix given is \[(I + 2A)^{-1} = \begin{pmatrix} -1 & 2 \ 4 & 5 \end{pmatrix}\] which implies that\[I + 2A = \left( \begin{pmatrix} -1 & 2 \ 4 & 5 \end{pmatrix}\right)^{-1}.\] We need to find this inverse.
02

Finding the Inverse of the Given Matrix

The formula for the inverse of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is \[ \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}. \] For our matrix \( \begin{pmatrix} -1 & 2 \ 4 & 5 \end{pmatrix} \), the determinant is \[ (-1)(5) - (2)(4) = -5 - 8 = -13. \] Therefore, the inverse is \[ \frac{1}{-13} \begin{pmatrix} 5 & -2 \ -4 & -1 \end{pmatrix} = \begin{pmatrix} -\frac{5}{13} & \frac{2}{13} \ \frac{4}{13} & \frac{1}{13} \end{pmatrix}. \]
03

Identifying Matrix I + 2A

Now that we have the inverse, we can say \[I + 2A = \begin{pmatrix} -\frac{5}{13} & \frac{2}{13} \ \frac{4}{13} & \frac{1}{13} \end{pmatrix}.\] This represents the sum of the identity matrix \(I\) and double the matrix \(A\).
04

Separating the Identity Matrix

The identity matrix \(I\) in 2x2 form is \[ \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}. \] Subtract \(I\) from both sides of the equation to isolate \(2A\): \[2A = \begin{pmatrix} -\frac{5}{13} & \frac{2}{13} \ \frac{4}{13} & \frac{1}{13} \end{pmatrix} - \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} -\frac{5}{13} - 1 & \frac{2}{13} \ \frac{4}{13} & \frac{1}{13} - 1 \end{pmatrix} = \begin{pmatrix} -\frac{18}{13} & \frac{2}{13} \ \frac{4}{13} & -\frac{12}{13} \end{pmatrix}. \]
05

Solving for A

Now, divide the result by 2 to solve for \(A\): \[A = \frac{1}{2} \begin{pmatrix} -\frac{18}{13} & \frac{2}{13} \ \frac{4}{13} & -\frac{12}{13} \end{pmatrix} = \begin{pmatrix} -\frac{9}{13} & \frac{1}{13} \ \frac{2}{13} & -\frac{6}{13} \end{pmatrix}. \] \(A\) is now isolated.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse of a Matrix
The inverse of a matrix, much like the reciprocal of a number, "undoes" the effect of the matrix when multiplied together.
The result of multiplying a matrix by its inverse is the identity matrix, denoted as "I". The identity matrix acts like the number 1 in matrix algebra, leaving the matrix unchanged when multiplied.

For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), there is a specific formula for its inverse:
  • The determinant, \( ad - bc \), is crucial; it needs to be non-zero for the inverse to exist.
  • The inverse is calculated as \( \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \).
Understanding the inverse allows us to solve matrix equations by transforming them into simpler forms. It is particularly useful in systems of equations and other applications.
2x2 Matrix
A 2x2 matrix is a simple form of a matrix which has two rows and two columns. It is fundamental to understand matrices of this size as they form the base for more complex matrices.

The general structure of a 2x2 matrix is represented as \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \).
This compact form allows for concise computations in linear algebra.

In matrix algebra:
  • Each element in the matrix can represent different variables or constants in mathematical expressions.
  • Operations such as addition, multiplication, and finding an inverse are applicable.
Being able to compute such operations with a 2x2 matrix provides a foundational skill set for higher-dimensional matrix work.
Determinant
The determinant is a special number calculated from a square matrix. For a 2x2 matrix, it is the difference between the products of its diagonals.

In the 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is found by:\[ ad - bc \]
This calculation is straightforward yet pivotal.
  • If the determinant is zero, the matrix has no inverse and is considered singular.
  • A non-zero determinant indicates that the matrix is invertible and its rows and columns are linearly independent.
The determinant also provides critical insights into the nature of the matrix, such as its volume distortion properties in transformations.

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Most popular questions from this chapter

Fill in the missing entries (marked with \(x\) ) to produce a skew-symmetric matrix. $$A=\left[\begin{array}{ccc} x & x & 4 \\ 0 & x & x \\ x & -1 & x \end{array}\right]$$

How many \(3 \times 3\) matrices \(A\) can you find such that $$ A\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} x+y \\ x-y \\ 0 \end{array}\right] $$ for all choices of \(x, y,\) and \(z ?\) One; namely, \(A=\left[\begin{array}{ccc}1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & 0\end{array}\right]\)

Show that $$ A=\left[\begin{array}{lllll} 0 & a & 0 & 0 & 0 \\ b & 0 & c & 0 & 0 \\ 0 & d & 0 & e & 0 \\ 0 & 0 & f & 0 & g \\ 0 & 0 & 0 & h & 0 \end{array}\right] $$ is not invertible for any values of the entries.

Use the inversion algorithm to find the inverse of the given matrix, if the inverse exists. $$\left[\begin{array}{rrr} 3 & 4 & -1 \\ 1 & 0 & 3 \\ 2 & 5 & -4 \end{array}\right]$$

Let \(A=\left[a_{i j}\right]\) be an \(n \times n\) matrix. Determine whether \(A\) is symmetric. (a) \(a_{i j}=i^{2}+j^{2}\) (b) \(a_{i j}=i^{2}-j^{2}\) (c) \(a_{i j}=2 i+2 j\) (d) \(a_{i j}=2 i^{2}+2 j^{3}\)
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