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Chapter 9: Problem 59

Let \(A=\left[\begin{array}{ll}1 & 1 \\ 1 & 3\end{array}\right] .\) Find \(A^{-1}\)

Short Answer

Expert verified
The inverse of matrix \( A \) is \( A^{-1} = \begin{pmatrix} \frac{3}{2} & -\frac{1}{2} \ -\frac{1}{2} & \frac{1}{2} \end{pmatrix}. \)

Step by step solution

01

- Calculate the Determinant

For a 2x2 matrix \[ A = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \] the determinant is calculated as \[ \text{det}(A) = ad - bc. \] For the given matrix \(A = \begin{pmatrix} 1 & 1 \ 1 & 3 \end{pmatrix}, \) the determinant \(\text{det}(A)\) is \[ 1 \times 3 - 1 \times 1 = 2. \]
02

- Check if the Matrix is Invertible

A matrix is invertible if its determinant is not zero. Since \(\text{det}(A) = 2 eq 0\), matrix \( A \) is invertible.
03

- Find the Adjugate Matrix

For a 2x2 matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \) the adjugate (or adjoint) matrix is \[ \text{adj}(A) = \begin{pmatrix} d & -b \ -c & a \end{pmatrix}. \] For the given matrix \( A = \begin{pmatrix} 1 & 1 \ 1 & 3 \end{pmatrix}, \) the adjugate matrix is \[ \text{adj}(A) = \begin{pmatrix} 3 & -1 \ -1 & 1 \end{pmatrix}. \]
04

- Calculate the Inverse Matrix

The inverse of matrix \( A \) is calculated using the formula \[ A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A). \] Substituting the values we have: \[ A^{-1} = \frac{1}{2} \begin{pmatrix} 3 & -1 \ -1 & 1 \end{pmatrix} = \begin{pmatrix} \frac{3}{2} & -\frac{1}{2} \ -\frac{1}{2} & \frac{1}{2} \end{pmatrix}. \]

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Key Concepts

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

determinant
The determinant is a scalar value that can be computed from a square matrix. It summarizes some important properties of the matrix. For a 2x2 matrix, it is calculated as follows:

For a matrix \(\begin{pmatrix} a & b \ c & d \)\right), the determinant is given by:

\[ \text{det}(A) = ad - bc. \]

Using the matrix from our example:

\[ A = \begin{pmatrix} 1 & 1 \ 1 & 3 \ \right), \]

we compute:

\[ \text{det}(A) = 1 \times 3 - 1 \times 1 = 2. \]

Since the determinant is not zero, this matrix is invertible.
adjugate matrix
The adjugate (or adjoint) matrix is used in finding the inverse of a matrix. For a 2x2 matrix,

\( A = \begin{pmatrix} a & b \ c & d \ \), \ the adjugate is just a rearrangement of these elements:

\[ \text{adj}(A) = \begin{pmatrix} d & -b \ -c & a \ ). \]

In our example:

\( A = \begin{pmatrix} 1 & 1 \ 1 & 3 \ \right), \

the adjugate is:

\[ \text{adj}(A) = \begin{pmatrix} 3 & -1 \ -1 & 1 \ \). \]

This rearrangement allows us to use it in the formula to compute the inverse matrix.
inverse matrix
The inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. For finding the inverse of our 2x2 matrix, we use:

\[ A^{-1} = \frac{1}{\text{det}(A)} \ \text{adj}(A). \]

From our computed determinant and adjugate:

The determinant is 2, and the adjugate matrix is:

\(\begin{pmatrix} 3 & -1 \ -1 & 1 \ \right) \),\

Thus:

\(A^{-1} = \frac{1}{2} \ \begin{pmatrix} 3 & -1 \ -1 & 1 \)) = \begin{pmatrix} \frac{3}{2} & -\frac{1}{2} \ -\frac{1}{2} & \frac{1}{2} \ \right). \

This is the inverse matrix for our example.
2x2 matrix
A 2x2 matrix has two rows and two columns. It is one of the simplest forms of matrices and is foundational in linear algebra. Here's a given matrix expressed generally:

\( A = \begin{pmatrix} a & b \ c & d \ \), \

In this structure:
  • Elements \(a \ and \ d \) are on the main diagonal.
  • Elements \(b \ and \ c \) are off the diagonal.


Two primary operations with 2x2 matrices are calculating determinants and finding inverses, both of which we've demonstrated.

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Most popular questions from this chapter

Solve each problem, using a system of three equations in three unknowns and Cramer鈥檚 rule. Bennie鈥檚 Coins Bennie emptied his pocket of 49 coins to pay for his $5.50 lunch. He used only nickels, dimes, and quarters, and the total number of dimes and quarters was one more than the number of nickels. How many of each type of coin did he use?

Write a matrix equation of the form \(A X=B\) that corresponds to each system of equations. $$\begin{aligned}&x-y=-7\\\&x+2 y=8\end{aligned}$$

Find the inverse of each matrix \(A\) if possible. Check that \(A A^{-1}=I\) and \(A^{-1} A=I .\) See the procedure for finding \(A^{-1}\) $ $$\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 0 \\ 1 & 3 & 0\end{array}\right]$$

Prove each of the following statements for any \(3 \times 3\) matrix \(A\). If all entries in any row or column of \(A\) are zero, then \(|A|=0\).

Solve each system of equations by using \(A^{-1} .\) Note that the matrix of coefficients in each system is a matrix. \(\begin{aligned} x+2 z &=-4 \\ 2 y &=6 \\ x+3 y &=7 \end{aligned}\)
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