# If the matrix A is such that $A = \left[ {\begin{array}{*{20}{c}} 3\\ { - 2}\\ 4 \end{array}} \right]\left[ {2\;\;\;5\;\;\;-2} \right]$ Then the det

19 views

closed

If the matrix A is such that

$A = \left[ {\begin{array}{*{20}{c}} 3\\ { - 2}\\ 4 \end{array}} \right]\left[ {2\;\;\;5\;\;\;-2} \right]$

Then the determinant of A is equal to ____

by (30.0k points)
selected

Correct Answer - Option 1 : 0

Concept:

The determinant of Matrix  $M = {\left[ {\begin{array}{*{20}{c}} a\\ b\\ :\\ d \end{array}} \right]_{n \times 1}}{\left[ {p\;\;\;q\;\;\;..\;\;s} \right]_{1 \times n\;}}$  is 0, when two of either row or column will be the same.

Calculation:

$A = \left[ {\begin{array}{*{20}{c}} 3\\ { - 2}\\ 4 \end{array}} \right]\left[ {2\;\;\;5\;\;\;-2} \right]$

$A = \;{\left[ {\begin{array}{*{20}{c}} 6&{15}&{-6}\\ { - 4}&{ - 10}&{ -4}\\ 8&{20}&{-8} \end{array}} \right]_{3 \times 3}}$

$\left| A \right| = \;\left| {\begin{array}{*{20}{c}} 6&{15}&{-6}\\ { - 4}&{ -10}&{ 4}\\ 8&{20}&{-8} \end{array}} \right|$

$\left| A \right| = \;3 \times - 2 \times 4\left| {\begin{array}{*{20}{c}} 2&5&- 2\\ 2&5&- 2\\ 2&5&- 2 \end{array}} \right|$

Since 2 rows are identical determinant is 0

$\left| A \right| = \;2 \times - 4 \times 7 \times 0 = 0$

The determinant of A is equal to 0