Thread: Matrices

Hi Thank you all very much for your answers to my previous questions, really appreciate them. Unfortunately, I have one more..

Find the values of a for which the system of equations has a unique solution.
3x-ay+2z = 4
x+2y-3z = 1
-x-y+z = 12

In matrix form,
\\1 & 2 & -3 \\-1 & -1 & 1 \end{bmatrix}\begin{bmatrix} x \\y \\z \end{bmatrix}" alt="\begin{bmatrix} 3 & -a & 2 \\1 & 2 & -3 \\-1 & -1 & 1 \end{bmatrix}\begin{bmatrix} x \\y \\z \end{bmatrix}" />
= \\1\\12 \end{bmatrix}" alt="\begin{bmatrix} 4 \\1\\12 \end{bmatrix}" />

I know I am supposed to solve it such that the determinant is not zero to get a unique solution..I know how to get the determinant of 3 by 3 matrices, I got a = -0.5 (or is it 0.5? ( I was confused with the negative signs)) for determinant =0. However, I was not taught the inverse of a 3 by 3 matrix, I was told to key it into the calculator to which I can do, but not here, since there is an unknown a..I tried substituting with a=-0.5 but I goe fractional entries, and I do not think I am doing it the right way anyway..

Please help me!
I have done such questions before but they were only asking for the inverse of a two by two matrix, and I could easily find the value of the unknown etc..

the determinant we want is:



for a 3x3 matrix, the easiest method to employ is the "rule of sarrus" (form 3 triples product by "wrap-around diagonals" from upper-left to lower right, and sum these, form three more triple products from "wrap around diagonals" from upper-right to lower-left, subtract these from the first sum).

in the matrix at hand that gives us:





if the determinant is non-zero, it means the matrix IS invertible, if we call our matrix A, then the unique solution is A^-1(4,1,12).

the determinant will be non-zero whenever:

that is, when

note that what happens when a = -1/2, is the first column is: twice the second column plus the third. this means that A is of rank 2 (since the 2nd and 3rd columns are clearly LI), and that the image (column space) of A is:

span({(1/2,2,-1),(2,-3,1)}). in this particular case, when a = -1/2, the system has NO solution, as (4,1,12) is not IN the column space.

since the problem does not ask you to specifically give a solution when a ≠ -1/2, this is all that needs be done.

Thank you so much! I get why the answer is 'no solutions'!

Could I trouble you to elaborate more about the part where you identified A is rank 2, I don't remember learning that in school, and we were never taught to look at the relationship of entries within a matrix in such an intricate way!