Question

If Δ is the value of the determinant

\(\left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right|\)

then what is the value of the following determinant?

\(\left| {\begin{array}{*{20}{c}} {{pa_1}}&{{b_1}}&{{qc_1}}\\ {{pa_2}}&{{b_2}}&{{qc_2}}\\ {{pa_3}}&{{b_3}}&{{qc_3}} \end{array}} \right|\)

(p ≠ 0 or 1, q ≠ 0 or 1)

1. pΔ
2. qΔ
3. (p + q)Δ
4. pqΔ

Answer 1

Correct Answer - Option 4 : pqΔ

Concept:

Using Scalar Multiple Property:

If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.

Calculation:

Δ = \(\left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right|\)

Column 1 of all elements is multiplied by p then given matrices is also multiplied by p

pΔ = \(​​\)\(\left| {\begin{array}{*{20}{c}} {{pa_1}}&{{b_1}}&{{c_1}}\\ {{pa_2}}&{{b_2}}&{{c_2}}\\ {{pa_3}}&{{b_3}}&{{c_3}} \end{array}} \right|\)

Again column 3 of all elements is multiplied by q then given matrices is also multiplied by q

pqΔ = \(\left| {\begin{array}{*{20}{c}} {{pa_1}}&{{b_1}}&{{qc_1}}\\ {{pa_2}}&{{b_2}}&{{qc_2}}\\ {{pa_3}}&{{b_3}}&{{qc_3}} \end{array}} \right|\)