The inverse of the matrix - 2 0 -1- 5 1 0- 0 1 3 - is

The correct option is D None of theseLet A = #160;[ 2 amp; 0 amp; 1; 5 amp; 1 amp; 0; 0 amp; 1 amp; 3 ] We known than A = IA . There ...

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Standard XII

Mathematics

Inverse of a Matrix

The inverse o...

Question

The inverse of the matrix $⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$ is

⎡ ⎢ ⎣ \begin{matrix} 3 & 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦

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⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦

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⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & 2 & 2 \end{matrix} ⎤ ⎥ ⎦

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None of these

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Solution

The correct option is D None of these
Let $A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$
We known than $A = I A$ . Therefore,
$⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$
Applying $R_{1} \to \frac{1}{2} R_{1}$ , we have
$⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{1}{2} 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ A$
Applying $R_{2} \to R_{2} - 5 R_{1}$ , we have
$⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{1}{2} 0 & 1 & \frac{5}{2} 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} & 0 & 0 - \frac{5}{2} & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
Applying $R_{3} \to R_{3} - R_{2}$ , we have
$⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{1}{2} 0 & 1 & \frac{5}{2} 0 & 0 & \frac{1}{2} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} & 0 & 0 - \frac{5}{2} & 1 & 0 \frac{5}{2} & - 1 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ A$
Applying $R_{3} \to 2 R_{3}$ , we have
$⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & - \frac{1}{2} 0 & 1 & \frac{5}{2} 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} & 0 & 0 - \frac{5}{2} & 1 & 0 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ A$
Applying $R_{1} \to R_{1} + \frac{1}{2} R_{3}$ and $R_{2} \to R_{2} - \frac{5}{2} R_{3}$ , we have
$⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦ A$
$∴ A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦$

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Similar questions

A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦

and

B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & - 2 - 1 & 3 & 0 0 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦

, then

{(A B)}^{- 1} = ?

Q. If

A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦

and

B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & - 2 - 1 & 3 & 0 0 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦

, find

{(A B)}^{- 1}

Using elementary transformations, find the inverse of matrix $⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$ , if it exists.

The sum of a number and its reciprocal is $\frac{125}{22}$ The number is

A. $\frac{2}{11}$

B. $\frac{1}{11}$

C. $\frac{3}{11}$

N o n e o f t h e s e .

Check whether following statement is true or false.

156

as a product of its prime factors is

2^{2} \times 3 \times 13

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The inverse of the matrix ⎡⎢⎣20−1510013⎤⎥⎦ is

The inverse of the matrix $⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & - 1 5 & 1 & 0 0 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦$ is