Question

Find the value of k for which the matrix {{-2,7,-18},{4,6,-4},{-2,1,k}} has rank 2

Answer 1

| -2 7 -18 |

|  4 6   -4 | = -2(6k+4)-7(4k-8)-18(4+12)=-12k-8-28k+56-288=-40k-240.

| -2 1    k |

-40k-240=0 when k=-6.

When the determinant of a 3 by 3 matrix is zero it implies that there is some dependence between rows and columns. If k=-6 the matrix becomes:

⎛ -2 7 -18 ⎞

⎜  4 6   -4 ⎟

⎝ -2 1   -6 ⎠

R1=R1-R3

⎛  0 6 -12 ⎞

⎜  4 6   -4 ⎟

⎝ -2 1   -6 ⎠

R2=R2/2

⎛  0 6 -12 ⎞

⎜  2 3   -2 ⎟

⎝ -2 1   -6 ⎠

R3=R3+R2

⎛  0 6 -12 ⎞

⎜  2 3   -2 ⎟

⎝  0 4   -8 ⎠

R1=R1/6; R3=R3/4

⎛  0 1   -2 ⎞

⎜  2 3   -2 ⎟

⎝  0 1   -2 ⎠

R3=R3-R2

⎛  0 1  -2 ⎞

⎜  2 3  -2 ⎟

⎝  0 0   0 ⎠

There are only two non-zero rows so the rank is 2 when k=-6.