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 $|A| 鈮� 0$ .

- 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) 鈭� (10) (1) \\ = 8 鈭� 10 \\ = 鈭� 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 $2 脳 2$ 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 $a d - b c 鈮� 0$ .

- Description of step.

The inverse of the given matrix is given by:

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

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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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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.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Theoretical and Mathematical Physics

Calculus

Geometry

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Study anywhere. Anytime. Across all devices.

- Description of step.

- Find the determinant of the given matrix.

- Description of step.

- Description of step.