Chapter 4: Q45. (page 201)

What is the inverse of $[\begin{matrix} 4 & 1 \\ 10 & 2 \end{matrix}]$ ?
A $[\begin{matrix} - 1 & \frac{1}{2} \\ 5 & - 2 \end{matrix}]$ B $[\begin{matrix} 2 & - 1 \\ - 10 & 4 \end{matrix}]$ C $[\begin{matrix} 1 & 5 \\ \frac{1}{2} & 2 \end{matrix}]$ D $[\begin{matrix} 1 & 5 \\ \frac{1}{2} & 2 \end{matrix}]$

Short Answer

Expert verified

The inverse of the given matrix is: $A [\begin{matrix} - 1 & \frac{1}{2} \\ 5 & - 2 \end{matrix}]$

Step by step solution

- Description of step.

A square matrix $A$ does not have its inverse if $|A| = 0$ .

A square matrix $A$ have its inverse if $|A| \neq 0$ .

- Find the determinant of the given matrix.

The determinant of the given matrix is:

\begin{matrix} |\begin{matrix} 4 & 1 \\ 10 & 2 \end{matrix}| = (4) (2) - (10) (1) \\ = 8 - 10 \\ = - 2 \end{matrix}

As the determinant of the given matrix is not equal to zero, therefore the inverse of the given matrix exists.

- Description of step.

The inverse of $2 \times 2$ matrix $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ is given by $A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$

and $a d - b c \neq 0$ .

- Description of step.

The inverse of the given matrix is given by:

\begin{matrix} \frac{1}{(4) (2) - (10) (1)} [\begin{matrix} 2 & - (1) \\ - (10) & 4 \end{matrix}] = \frac{1}{8 - 10} [\begin{matrix} 2 & - 1 \\ - 10 & 4 \end{matrix}] \\ = \frac{1}{- 2} [\begin{matrix} 2 & - 1 \\ - 10 & 4 \end{matrix}] \\ = [\begin{matrix} \frac{2}{- 2} & \frac{- 1}{- 2} \\ \frac{- 10}{- 2} & \frac{4}{- 2} \end{matrix}] \\ = [\begin{matrix} - 1 & \frac{1}{2} \\ 5 & - 2 \end{matrix}] \end{matrix}

Therefore, the inverse of the given matrix is $[\begin{matrix} - 1 & \frac{1}{2} \\ 5 & - 2 \end{matrix}]$

Therefore, the option A is correct.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

- Description of step.

- Find the determinant of the given matrix.

- Description of step.

- Description of step.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Applied Mathematics

Calculus

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

What is the inverse of41102?A-1125-2 B2-1-104C15122 D15122

Short Answer

Step by step solution

­- Description of step.

­- Find the determinant of the given matrix.

­- Description of step.

­- Description of step.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Applied Mathematics

Calculus

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

- Description of step.

- Find the determinant of the given matrix.

- Description of step.

- Description of step.