SOLUTION: How does the rank of the following matrix depend on the value of t? (1,1,t) (1,t,1) (t,1,1)

Question 1167394: How does the rank of the following matrix depend on the value of t?
(1,1,t)
(1,t,1)
(t,1,1)

Answer by Resolver123(6) (Show Source): You can put this solution on YOUR website!
We are given the following 3x3 matrix:

Compute the determinant det(A):

.
Let , or .
Hence, det(A) = 0 if and only if t = 1 or t = -2.

Consider 3 cases:

Case 1: t ≠ 1 and t ≠ -2.
Then det(A) ≠ 0, and so the matrix is of full rank, that is, rank(A) = 3.

Case 2: t = 1
Then the matrix is:

All rows being identical means that there is only 1 linearly independent row. Hence, rank = 1.

Case 3: t = -2
Then we get the matrix:

Using the row operations and , we get the row equivalent matrix

Using the row operation , we finally get

This gives 2 linearly independent rows, and therefore, rank = 2.

Thus, the rank of the matrix depends on as follows:
* Rank = 3 if t ≠ 1 and t ≠ -2.
* Rank = 2 if t = -2, and
* Rank = 1 if t = 1.

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