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The question is:

How many times is the value of the determinant of 2x2 matrix increase when each element of the matrix is multiplied by:

a) 2

b) 3

c)-2

d)-3

e) k

But I didn't understand what the question is asking, or what should I do.

Please, overwatch needs me

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    Hint: matrix $A$ with first row $(a,b)$ and second row $(c,d)$, have determinant $det(A) = ad-bc$. – 2017-01-25

4 Answers 4

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The question could be reworded as follows:

If $A$ is a $2\times 2$ matrix and has determinant $d$, find the determinant (as a function of $d$) of the following matrices:

  • $2A$
  • $3A$
  • $-2A$
  • $-3A$
  • $kA$

To find the final answer you could either think of the way the determinant is defined, being multi-linear, or use the specific formula for the $2\times2$ determinant, which is $\det(A) = xw - yz$ if $A = \left(\begin{matrix}x&y\\z&w\end{matrix}\right)$

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Hint: If you multiply one row by $a$ then the dererminant is multiplies by $a$. Multiplying every element by $a$ corresponds to doing this twice.

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Hint: Determinant of $n\times n$ matrix increases $k^n$ times when each element of matrix is multiplied by $k$.

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The determinant of a 3 by 3 matrix is the sum of all possible products of one number from each row and each column, multiplied by either "1" or "-1". So it is a sum and difference of products of three numbers. If each of those numbers is multiplied by "a" then the product of three such numbers is multiplied by $a^3$.