Chapter 4: Q26. (page 199)

Find the inverse of each matrix, if it exists.26.
$[\begin{matrix} 4 & - 3 \\ 2 & 7 \end{matrix}]$

Short Answer

Expert verified

The inverse of the matrix is $[\begin{matrix} \frac{7}{34} & \frac{3}{34} \\ - \frac{1}{17} & \frac{2}{17} \end{matrix}]$

Step by step solution

- Define inverse of a matrix.

For the matrix A ,

$A = [\begin{matrix} a & b \\ c & d \end{matrix}]$

The inverse of matrix of the matrix A is:

$A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$

where $a d - b c \neq 0$ . $a d - b c$ is the determinant of the matrix A .

If $a d - b c = 0$ , the inverse of a matrix doesn not exist.

- Calculate the inverse.

Let be the matrix $[\begin{matrix} 4 & - 3 \\ 2 & 7 \end{matrix}]$

That is, $A = [\begin{matrix} 4 & - 3 \\ 2 & 7 \end{matrix}]$

Comparing with the standard form $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ $a = 4, b = - 3, c = 2, d = 7$

Then, $A^{- 1}$ is:

$\begin{matrix} A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}] \\ = \frac{1}{(4 \times 7) - (- 3 \times 2)} [\begin{matrix} 7 & - (- 3) \\ - 2 & 4 \end{matrix}] \\ = \frac{1}{28 + 6} [\begin{matrix} 7 & 3 \\ - 2 & 4 \end{matrix}] \\ = \frac{1}{34} [\begin{matrix} 7 & 3 \\ - 2 & 4 \end{matrix}] \end{matrix}$

Here, $\begin{matrix} a d - b c = 28 + 6 \\ = 34 \end{matrix}$

As $a d - b c \neq 0$ here, inverse exist.

Then, $A^{- 1}$ is:

$\begin{matrix} A^{- 1} = \frac{1}{34} [\begin{matrix} 7 & 3 \\ - 2 & 4 \end{matrix}] \\ = [\begin{matrix} \frac{7}{34} & \frac{3}{34} \\ - \frac{2}{34} & \frac{4}{34} \end{matrix}] \\ = [\begin{matrix} \frac{7}{34} & \frac{3}{34} \\ - \frac{1}{17} & \frac{2}{17} \end{matrix}] \end{matrix}$

- State the conclusion.

Therefore, the inverse of the matrix $[\begin{matrix} 4 & - 3 \\ 2 & 7 \end{matrix}]$ is $[\begin{matrix} \frac{7}{34} & \frac{3}{34} \\ - \frac{1}{17} & \frac{2}{17} \end{matrix}]$

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.

Find the inverse of each matrix, if it exists.26.
$[\begin{matrix} 4 & - 3 \\ 2 & 7 \end{matrix}]$

Short Answer

Step by step solution

- Define inverse of a matrix.

- Calculate the inverse.

- State the conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Calculus

Pure Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Find the inverse of each matrix, if it exists.26.[4−327]

Short Answer

Step by step solution

- Define inverse of a matrix.

- Calculate the inverse.

- State the conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Calculus

Pure Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Find the inverse of each matrix, if it exists.26.
$[\begin{matrix} 4 & - 3 \\ 2 & 7 \end{matrix}]$