Perform the following operations in the matrix $⎡ ⎢ ⎣ \begin{matrix} 3 & 4 & 45 7 & 9 & 105 13 & 2 & 195 \end{matrix} ⎤ ⎥ ⎦$
(i) Add the third row to the second row
(ii) Subtract the third column from the first column

The determinant of the resultant matrix is
0

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Solution

The correct option is A 0
Let
|A|= $∣ ∣ ∣ ∣ \begin{matrix} 3 & 4 & 45 7 & 9 & 105 13 & 2 & 195 \end{matrix} ∣ ∣ ∣ ∣$
=15 $∣ ∣ ∣ ∣ \begin{matrix} 3 & 4 & 3 7 & 9 & 7 13 & 2 & 13 \end{matrix} ∣ ∣ ∣ ∣$ =15 $\times$ 0=0

Now we know that the value of the determinant remains same after performing the operations $R_{i} + K R_{j}$ and $C_{i} + K C_{j}$ (Linear transform does not change value of determinant)

So, the determinant of resultant matrix = 0.

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a_{i j}

a_{i j} = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 0, & i = j 1, & i > j - 1, & i < j \end{matrix}

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