Question

The determinant of  \(\begin{vmatrix} p & q & r \\ x & y & z \\ a & b& c \end{vmatrix}\) = 5, then find determinant of \(\begin{vmatrix} kp & kq &k r \\ x & y & z \\ a & b& c \end{vmatrix}\) where k is any scalar ?

1. 5 
2. 5k
3. 5k²
4. 5k³

Answer 1

Correct Answer - Option 2 : 5k

Concept:

Properties of Determinant:

Any scalar multiplied to a row or a column can be taken out, e.g.

\(\begin{vmatrix} ka_{11} &ka_{12} & ka_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix}\) = k\(\begin{vmatrix} a_{11} &a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix}\) 

= k[a11(a22 a33 - a23 a32) - a12(a21 a33 - a23 a31) + a13(a21 a32 - a22 a31)]

 

Calculation

Given \(\begin{vmatrix} p & q & r \\ x & y & z \\ a & b& c \end{vmatrix}\) = 5

Now,

\(\begin{vmatrix} kp & kq &k r \\ x & y & z \\ a & b& c \end{vmatrix}\) =  k\(\begin{vmatrix} p & q & r \\ x & y & z \\ a & b& c \end{vmatrix}\)

∴ 5k