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

Question

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.

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Answer 1

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.

Answer 2

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

Answer 3

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 $A = \left(\begin{matrix}x&y\\z&w\end{matrix}\right)$

Answer 4

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 $a^3$.