Question

Is Ax = B consistent or inconsistent ?

Answer 1

The system is consistent. See explanation.

Explanation:

The system of equations `Ax=B` is consistent if `detA!=0`. Here we have:

`detA=|(1,0,6),(-2,8,5),(6,5,1)|=`

`=1*8*1+0*5*1+6*(-2)*(-2)-6*8*1-5*(-2)*1-1*0*(-2)=`

`8+24-48+10+0=-6`

`detA!=0`, so the system is consistent.

Answer 2

The system is consistent.

Here's an approach without explicitly using determinants.

Explanation:

The augmented matrix `(A|b)` is given by
`((1,0,6,|,1),(-2,8,5,|,3),(1,-2,1,|,-1))`

`R_2 -> R_2+2R_1, R_3 -> R_3-R_1`:
`((1,0,6,|,1),(0,8,17,|,5),(0,-2,-5,|,-2))`

`R_2 -> R_2/8`
`((1,0,6,|,1),(0,1,17/8,|,5/8),(0,-2,-5,|,-2))`

`R_3 -> R_3+2R_2`
`((1,0,6,|,1),(0,1,17/8,|,5/8),(0,0,-3/4,|,-3/4))`

`R_3 -> R_3/(-3/4)`
`((1,0,6,|,1),(0,1,17/8,|,5/8),(0,0,1,|,1))`

Both the augmented matrix `(A|b)` and the coefficient matrix `A` have a rank of 3 - so the system is consistent.

As an added advantage, this method gives a direct way of finding the solution as well.
`x_3 = 1`,
`x_2+17/8x_3=5/8 implies x_2 = -3/2`
`x_1+6x_3 = 1 implies x_1 = -5`