Question 6
The augmented matrix shown is produced when a Gaussian elimination technique is used to solve a certain system of equations with three variables. Each equation represents a plane.
$$\left[\begin{array}{ccc|c}1 & -1 & 2 & 4\\0 & 1 & -3 & 1\\0 & 0 & 0 & 5\end{array}\right]$$
Which statement is true?
That’s it. Step through the hints below to see why.
Not quite. You read row 3 as $z=5$, ignoring the zero coefficients and treating the 5 as the value of $z$.
Not quite. You treated row 3 as a zero row ($0=0$), ignoring the non-zero constant 5.
Not quite. The first half is right, but “no solution” does not mean the planes are parallel. Three planes are all parallel only if all three normal vectors are parallel, which would make the rank of the coefficient matrix 1. Row operations do not change that rank, and here it is 2.
Choose any point $(x,y,z)$. Each line shows the left-hand side of one row (dot) against its right-hand side (black bar).
Rows 1 and 2 both hold at this point, but row 3 still gives 0, not 5. So this point is not a solution either.
Row 3 gives 0, never 5, whatever values of $x$, $y$ and $z$ you choose. No point can make it true.
Rewrite each row as an equation.
$$\begin{aligned}x-y+2z&=4\\y-3z&=1\\0x+0y+0z&=5\end{aligned}$$
An augmented matrix is shorthand for a system. Each row holds the coefficients of $x$, $y$ and $z$, and the number after the bar is the right-hand side.
Simplify row 3. For any values of $x$, $y$ and $z$, the left-hand side is $0x+0y+0z=0$. So row 3 says
$$0=5,$$
which is false. So the system has no solution.
Row 3 is the row that decides the type of solution, so look at it first.
A solution must make every equation true at the same time. No choice of $(x,y,z)$ can make row 3 true, so no point satisfies all three equations.
Translate “no solution” into geometry. The original system also has no solution, so there is no point common to all three planes.
Each original equation is a plane, and a solution of the system is a point lying on every plane. Row operations do not change the solution set.
Choose the option that is fully true. (A) states exactly what you found: no solution, and no point common to all three planes. (D) claims more than that, so check it separately. Pick (D) above to see why it fails.
Both (A) and (D) say “no solution”, so check the geometric half of each.
Row 3 says $0=5$, which is false for every $(x,y,z)$. So the system has no solution, and there is no point common to all three planes.
Only mark this done when you could do it without help. Reading the solution does not count.
A Tangent Tuition mock exam question. It is not a past QCAA question, and Tangent Tuition is not affiliated with the QCAA.
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