Find all values of a for which the matrices of S= [1_0_0_0],[2_0_1_0],[2_a_3_0],[a_1_2_a-2] is linearly dependent.
first row reduce and switch the rows a bit to yield S= [1,0,0,0],[0,a,0,0],[0,0,1,0],[0,1,0,a-1]. I presume you already know that if the rows of S are linearly dependent, then that must mean one of the rows is a linear combination (or scalar multiple) of the others. It becomes apparent that S is linearly dependent when a=0 or a=1. When a=0 the 2nd row is all zeroes, and a row of all zeroes is just a linear combination of the other rows multiplied by zero. When a=1 the 4th and 2nd row are scalar multiples of each other. (4th row times a = 2nd row)
I made a small mistake in the previous question:
by substituting a for 1, row 4 = row 2 = [0,1,0,0]. technically they are still scalar multiple of each other (multiply by 1).(4th row times a = 2nd row)
Sorry but I'm having trouble solving that other matrix. I'm too short on time to devote too much energy on a single question. It's been a while since I've done questions like this. Btw do you know if it's ok to divide a row by a? As far as I know dividing by a adds the restriction that a=/=0, and by introducing a restriction you are changing the solution of the matrix, however in a situation where you have [0,0,0,a|0] (an augmented matrix), it's ok to divide by a since a=/=0 in the first place. I'm not sure if this rule applies more generally to matrices like this though?