Function defined by Sum of Matrix

[siegfried2911]

Hi. I encountered a problem involving matrices.

What is the value of x when f(x) = 15 for a funtion defined by f(x) = [MATH] [(-3x,2),(x,-1)] + [(2x,5),(-4,-3)] [/MATH] . The choices for the answer are 3, 10, 1 and 5.

I have no idea how to solve these kinds of problems. Will it involve transformations?

 

[HallsofIvy]

This problem, as posted, makes no sense! The function given maps a number, x, to, essentially, a 2 by 2 matrix, \(\displaystyle f(x)= \begin{bmatrix} -3x & 2 \\ x & -1\end{bmatrix}+ \begin{bmatrix}2x & 5 \\ -4 & -3 \end{bmatrix}= \begin{bmatrix}-x & 7 \\ x- 4 & -4\end{bmatrix}\). There is NO value of x such that f(x) is a single number, 15.

 

[siegfried2911]

True. That is also what I thought of. At first I thought I misread it, until I saw the questionnaire. Maybe some portion of the correct question was mistakenly deleted?

 

[Deleted member 4993]

This problem, as posted, makes no sense! The function given maps a number, x, to, essentially, a 2 by 2 matrix, \(\displaystyle f(x)= \begin{bmatrix} -3x & 2 \\ x & -1\end{bmatrix}+ \begin{bmatrix}2x & 5 \\ -4 & -3 \end{bmatrix}= \begin{bmatrix}-x & 7 \\ x- 4 & -4\end{bmatrix}\). There is NO value of x such that f(x) is a single number, 15.

Could it be that "[]" meant "determinant"?

 

[siegfried2911]

Could it be that "[]" meant "determinant"?

I also tried that. The answer will be 3. Though I am not sure if that's the case.

 

[pka]

Could it be that "[]" meant "determinant"?

Yes indeed, determinant.
SEE HERE

 

[Deleted member 4993]

I also tried that. The answer will be 3. Though I am not sure if that's the case.

Please share your work.

 

[Steven G]

I'd bet all my marbles on it being determinant.

 

[Deleted member 4993]

I'd bet all my marbles on it being determinant.

I thought you had lost those ... sometime ago .....

 

[Steven G]

I thought you had lost those ... sometime ago .....

I worded my comment just so you had something to say!

 

[bZNyQ7C2]

The notation here is strange to me. Maybe some other people aren't used to it either. I use absolute value bars around a matrix to indicate the determinant. I've also seen "the determinant of matrix A" written as "Det(A)."

Here's how I would write this, assuming I understand the problem right ...

\(\displaystyle \left| {\left[ {\begin{array}{*{20}{c}} { - 3x}&2\\ x&{ - 1} \end{array}} \right]} \right| + \left| {\left[ {\begin{array}{*{20}{c}} {2x}&5\\ { - 4}&{ - 3} \end{array}} \right]} \right| = 15\)

In that form, I get \(\displaystyle x = 1\). The OP got \(\displaystyle x = 3\). I think he must've added the two matrices, then took the determinant of the sum. I think that's not correct. I think the determinant is not preserved under matrix addition. If anyone knows different, please correct me.