Question

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

Answer 1

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 - R₁ and R2 → R2 - R1

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

R₃ → R₃ -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