# The element in the ith row and the jth column of a determinant of third order is equal to 2(i + j). What is the value of the determinant?

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The element in the ith row and the jth column of a determinant of third order is equal to 2(i + j). What is the value of the determinant?

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Correct Answer - Option 1 : 0

Concept:

If all the elements of a row or column are zeroes, then the value of the determinant is zero.

To evaluate the determinant row or column operation is done.

Calculation:

The determinant can be written as,

$\begin{bmatrix} 2(1+1) & 2(1+2) &2(1+3) \\ 2(2+1) &2(2+2) & 2(2+3)\\ 2(3+1)&2(3+2) & 2(3+3) \end{bmatrix}$

$2\times \begin{bmatrix} 1+1 &1+2 & 1+3\\ 2+1 & 2+2 &2+3 \\ 2+1 & 3+2 & 3+3 \end{bmatrix}$

R3 → R3 - Rand R2 → R2 - R1

$2\times \begin{bmatrix} 1+1 &1+2 & 1+3\\ 1 & 1 &1 \\ 2 & 2 & 2 \end{bmatrix}$

R3 → R-2R2

$2\times \begin{bmatrix} 1+1 &1+2 & 1+3\\ 1 & 1 &1 \\ 0 & 0 & 0 \end{bmatrix}$

= 0

So, the value of the determinant is 0