Question

Let A = [a_ij] and B = [b_ij] be a square matrices of order 3 such that b_i1 = 2 a_i1, b_i2 = 3 a_i2 and b_i3 = 4 a_i3, i = 1, 2, 3
If |A| = 5, then |B| = _____________.

Answer 1

Given:
A and B are square matrices of order 3
b_i1 = 2 a_i1, b_i2 = 3 a_i2 and b_i3 = 4 a_i3, i = 1, 2, 3
|A| = 5


$$\begin{gathered} \mathrm{Let}A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right] \\ \mathrm{Since},b_{i1}=2a_{i1},b_{i2}=3a_{i2}\mathrm{and}{b}_{i3}=4a_{i3} \\ \mathrm{Therefore}, \\ {b}_{11}=2a_{11},b_{21}=2a_{21},b_{31}=2a_{31}, \\ {b}_{12}=3a_{12},b_{22}=3a_{22},b_{32}=3a_{32}, \\ {b}_{13}=4a_{13},b_{23}=4a_{23},b_{33}=4a_{33} \\ \left|B\right|=\left|\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right| \\ \Rightarrow \left|B\right|=\left|\begin{array}{ccc}2a_{11}& 3a_{12}& 4a_{13}\\ 2a_{21}& 3a_{22}& 4a_{23}\\ 2a_{31}& 3a_{32}& 4a_{33}\end{array}\right| \\ \mathrm{Taking}\mathrm{out}2,3\mathrm{and}4\mathrm{common}\mathrm{from}{C}_1,C_2\mathrm{and}{C}_3,\mathrm{respectively} \\ \Rightarrow \left|B\right|=2\times 3\times 4\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right| \\ \Rightarrow \left|B\right|=24\left|A\right| \\ \Rightarrow \left|B\right|=24\times 5\left(\because \left|A\right|=5\right) \\ \Rightarrow \left|B\right|=120 \end{gathered}$$

Hence, |B| = 120.