Chapter 4: Q8. (page 199)

Find the inverse of each matrix, if it exists.
$[\begin{matrix} - 5 & 1 \\ 7 & 4 \end{matrix}]$

Short Answer

Expert verified

The inverse of the given matrix is: $[\begin{matrix} \frac{- 4}{27} & \frac{1}{27} \\ \frac{7}{27} & \frac{5}{27} \end{matrix}]$

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} - 5 & 1 \\ 7 & 4 \end{matrix}| = (- 5) (4) - (7) (1) \\ = - 20 - 7 \\ = - 27 \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 $2 \times 2$ matrixlocalid="1648646974912" $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ is given by localid="1648646983272" $A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ and $a d - b c \neq 0$ .

- Description of step.

The inverse of the given matrix is given by:

$\begin{matrix} \frac{1}{(- 5) (4) - (7) (1)} [\begin{matrix} 4 & - (1) \\ - (7) & - 5 \end{matrix}] = \frac{1}{- 20 - 7} [\begin{matrix} 4 & - 1 \\ - 7 & - 5 \end{matrix}] \\ = \frac{1}{- 27} [\begin{matrix} 4 & - 1 \\ - 7 & - 5 \end{matrix}] \\ = [\begin{matrix} \frac{4}{- 27} & \frac{- 1}{- 27} \\ \frac{- 7}{- 27} & \frac{- 5}{- 27} \end{matrix}] \\ = [\begin{matrix} \frac{- 4}{27} & \frac{1}{27} \\ \frac{7}{27} & \frac{5}{27} \end{matrix}] \end{matrix}$

Therefore, the inverse of the given matrix is $[\begin{matrix} \frac{- 4}{27} & \frac{1}{27} \\ \frac{7}{27} & \frac{5}{27} \end{matrix}]$ .

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Find the inverse of each matrix, if it exists.
$[\begin{matrix} - 5 & 1 \\ 7 & 4 \end{matrix}]$

Short Answer

Step by step solution

- Description of step.

- Find the determinant of the given matrix.

- Description of step.

- Description of step.

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Find the inverse of each matrix, if it exists.-5174

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

Decision Maths

Logic and Functions

Statistics

Calculus

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Find the inverse of each matrix, if it exists.
$[\begin{matrix} - 5 & 1 \\ 7 & 4 \end{matrix}]$

- Description of step.

- Find the determinant of the given matrix.

- Description of step.

- Description of step.