Let M be a 3 3 matrix such that M((|gathered 1 0 0 gathered))|=((|gathered 1 2 3 gathered)…

Question

Let M be a $3	imes 3$ matrix such that $M((|\begin{gathered} 1 \ 0 \ 0 \end{gathered}))|=((|\begin{gathered} 1 \ 2 \ 3 \end{gathered}))|,M((|\begin{gathered} 0 \ 1 \ 0 \end{gathered}))|=((|\begin{gathered} 0 \ 1 \ 2 \end{gathered}))|$ and $M((|\begin{gathered} 0 \ 0 \ 1 \end{gathered}))|=((|\begin{gathered} -1 \ 1 \ 1 \end{gathered}))|$. If $M((|\begin{gathered} x \ y \ z \end{gathered}))|=((|\begin{gathered} 1 \ 7 \ 11 \end{gathered}))|$, then $x+y+z$ equals :

Accepted Answer

Since the images of the standard basis vectors are given, the columns of matrix $M$ are exactly these vectors.

So,

$M=((|\begin{gathered} 1 \quad  0 \quad  -1 \ 2 \quad  1 \quad  1 \ 3 \quad  2 \quad  1 \end{gathered}))|$

Now we are given

$M((|\begin{gathered} x \ y \ z \end{gathered}))|=((|\begin{gathered} 1 \ 7 \ 11 \end{gathered}))|$

So,

$((|\begin{gathered} 1 \quad  0 \quad  -1 \ 2 \quad  1 \quad  1 \ 3 \quad  2 \quad  1 \end{gathered}))|((|\begin{gathered} x \ y \ z \end{gathered}))|=((|\begin{gathered} 1 \ 7 \ 11 \end{gathered}))|$

This gives the system:

$x-z=1$

$2x+y+z=7$

$3x+2y+z=11$

Now solve step by step.

From the first equation,

$x=1+z$

Substitute into the second equation:

$2(1+z)+y+z=7$

$2+3z+y=7$

$y+3z=5\, \Rightarrow \, y=5-3z$

Now substitute $x=1+z$ and $y=5-3z$ into the third equation:

$3(1+z)+2(5-3z)+z=11$

$3+3z+10-6z+z=11$

$13-2z=11$

$-2z=-2$

$z=1$

Then,

$x=1+z=2$

and

$y=5-3z=2$

Therefore,

$x+y+z=2+2+1=5$

So the correct answer is:

$5$

Option B

Solutions · 1