If a 4x4 Matrix A with rows v1, v2, v3 and v4 has a det A = 3, then det [ 6v1+3v4; v2 ; v3; 4v1+8v4] = ???

Question

zarkon · Answer

$$A=\left(
  \begin{array}{c}
    v_1 \
    v_2 \
    v_3 \
    v_4 \
  \end{array}
\right)$$

with Det(A)=3

$$2r_1\rightarrow r_1$$

$$A_1=\left(
  \begin{array}{c}
    2v_1 \
    v_2 \
    v_3 \
    v_4 \
  \end{array}
\right)$$

$$r_4+r_1\rightarrow r_1$$

$$A_2=\left(
  \begin{array}{c}
    2v_1+v_4 \
    v_2 \
    v_3 \
    v_4 \
  \end{array}
\right)$$

$$6r_4\rightarrow r_4$$

$$A_3=\left(
  \begin{array}{c}
    2v_1+v_4 \
    v_2 \
    v_3 \
    6v_4 \
  \end{array}
\right)$$

$$2r_1+r_4\rightarrow r_4$$


$$A_4=\left(
  \begin{array}{c}
    2v_1+v_4 \
    v_2 \
    v_3 \
    4v_1+8v_4 \
  \end{array}
\right)$$

$$3r_1\rightarrow r_1$$

$$A_5=\left(
  \begin{array}{c}
    6v_1+3v_4 \
    v_2 \
    v_3 \
    4v_1+8v_4 \
  \end{array}
\right)$$

thus the $$Det(A_5)=3\cdot2\cdot6\cdot3=108$$

the 2,6,3 come from multiplying a row by a constant.