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Evaluation of a Determinant

Let A be a 3×...

Question

Let $A$ be a $3 \times 3$ matrix with $det (A) = 4.$ Let $R_{i}$ denote the $i^{th}$ row of $A .$ If a matrix $B$ is obtained by performing the operation $R_{2} \to 2 R_{2} + 5 R_{3}$ on $2 A,$ then $det (B)$ is equal to:

A

64

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B

16

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C

80

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D

128

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Solution

The correct option is A $64$
Let $A = ⎡ ⎢ ⎣ \begin{matrix} R_{11} & R_{12} & R_{13} R_{21} & R_{22} & R_{23} R_{31} & R_{32} & R_{33} \end{matrix} ⎤ ⎥ ⎦$

$\Rightarrow 2 A = ⎡ ⎢ ⎣ \begin{matrix} 2 R_{11} & 2 R_{12} & 2 R_{13} 2 R_{21} & 2 R_{22} & 2 R_{23} 2 R_{31} & 2 R_{32} & 2 R_{33} \end{matrix} ⎤ ⎥ ⎦$

Now, $R_{2} \to 2 R_{2} + 5 R_{3}$
So, $B = ⎡ ⎢ ⎣ \begin{matrix} 2 R_{11} & 2 R_{12} & 2 R_{13} 4 R_{21} + 10 R_{31} & 4 R_{22} + 10 R_{32} & 4 R_{23} + 10 R_{33} 2 R_{31} & 2 R_{32} & 2 R_{33} \end{matrix} ⎤ ⎥ ⎦$

$| B | = ∣ ∣ ∣ ∣ \begin{matrix} 2 R_{11} & 2 R_{12} & 2 R_{13} 4 R_{21} + 10 R_{31} & 4 R_{22} + 10 R_{32} & 4 R_{23} + 10 R_{33} 2 R_{31} & 2 R_{32} & 2 R_{33} \end{matrix} ∣ ∣ ∣ ∣$

$R_{2} \to R_{2} - 5 R_{3}$

$| B | = ∣ ∣ ∣ ∣ \begin{matrix} 2 R_{11} & 2 R_{12} & 2 R_{13} 4 R_{21} & 4 R_{22} & 4 R_{23} 2 R_{31} & 2 R_{32} & 2 R_{33} \end{matrix} ∣ ∣ ∣ ∣$

$∴ | B | = 2 \times 2 \times 4 ∣ ∣ ∣ ∣ \begin{matrix} R_{11} & R_{12} & R_{13} R_{21} & R_{22} & R_{23} R_{31} & R_{32} & R_{33} \end{matrix} ∣ ∣ ∣ ∣$
$= 16 \times 4$
$= 64$

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Similar questions

A

3 \times 3

det (A) = 4.

R_{i}

i^{t h}

A .

B

is obtained by performing the operation

R_{2} \to 2 R_{2} + 5 R_{3}

2 A,

det (B)

a

denote the element of the

i^{t h}

j^{t h}

3 \times 3

a_{i j} = - a_{j i}

for every i and j then this matrix is an -

A = [a_{i j}]

3 \times 3

matrix given by

a_{i j} = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (\frac{i + j}{2}) + \frac{| i - j |}{2}, & i f i \neq j \frac{i^{j} - (i . j)}{i^{2} + j^{2}}, & i f i = j \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭

a_{i j}

denotes element of

i^{t h}

j^{t h}

column of matrix A.If

A = p A^{2} - q A^{- 1} - r I

, then remainder when

(p + q + r)^{11}

is divided by 7 is

A = [a_{i j}]

3 \times 3

matrix given by

a_{i j} = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (\frac{i + j}{2}) + \frac{| i - j |}{2}, & i f i \neq j \frac{i^{j} - (i . j)}{i^{2} + j^{2}}, & i f i = j \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭

a_{i j}

denotes element of

i^{t h}

j^{t h}

column of matrix A.If

A = p A^{2} - q A^{- 1} - r I

, then remainder when

(p + q + r)^{11}

is divided by 7 is

I

3 \times 3

identity matrix and

P

be a matrix obtained by rearranging the columns of

I

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