Question

Alternatively, notice that we can find a and ? via the cofactors for $C_{11}$ and $C_{33}$, before taking the transpose of the cofactor matrix. $\alpha = COF_{11} = (+1)\begin{vmatrix} -3 & -3 \ 4 & -2 \end{vmatrix} = 18$ $\beta = COF_{33} = (+1)\begin{vmatrix} 2 & 4 \ 1 & -3 \end{vmatrix} = -10$ Alternatively, we can use the formula to find the inverse of a matrix. Marking: 4 marks for showing work, and 3 marks for the correct answer. (d) Using the inverse of the coefficients matrix $A^{-1}$ that you calculated in the previous File preview Ive for the (unknown) variables $x$, $y$, and $z$. [6 marks] Solution: Finally, we use the equation given in part (a) to find the solution: $Av = b$ $\Rightarrow v = A^{-1}b = Cb$ $\Rightarrow v = \frac{1}{58} \begin{pmatrix} 18 & 28 & 3 \ 8 & 6 & 11 \ -2 & -16 & -10 \end{pmatrix} \begin{pmatrix} 7 \ -11 \ 22 \end{pmatrix} = \begin{pmatrix} -2 \ 4 \ -1 \end{pmatrix}$ Marking: 3 marks for showing work, and 3 marks for the correct answer.

          Alternatively, notice that we can find a and ? via the cofactors for $C_{11}$ and $C_{33}$,
before taking the transpose of the cofactor matrix.
$\alpha = COF_{11} = (+1)\begin{vmatrix} -3 & -3 \ 4 & -2 \end{vmatrix} = 18$
$\beta = COF_{33} = (+1)\begin{vmatrix} 2 & 4 \ 1 & -3 \end{vmatrix} = -10$
Alternatively, we can use the formula to find the inverse of a matrix.
Marking: 4 marks for showing work, and 3 marks for the correct answer.
(d) Using the inverse of the coefficients matrix $A^{-1}$ that you calculated in the previous
File preview Ive for the (unknown) variables $x$, $y$, and $z$.
[6 marks]
Solution: Finally, we use the equation given in part (a) to find the solution:
$Av = b$
$\Rightarrow v = A^{-1}b = Cb$
$\Rightarrow v = \frac{1}{58} \begin{pmatrix} 18 & 28 & 3 \ 8 & 6 & 11 \ -2 & -16 & -10 \end{pmatrix} \begin{pmatrix} 7 \ -11 \ 22 \end{pmatrix} = \begin{pmatrix} -2 \ 4 \ -1 \end{pmatrix}$
Marking: 3 marks for showing work, and 3 marks for the correct answer.

Alternatively, notice that we can find a and ? via the cofactors for C11 and C33,
before taking the transpose of the cofactor matrix.
α = COF11 = (+1) -3 -3 4 -2 = 18
β = COF33 = (+1) 2 4 1 -3 = -10
Alternatively, we can use the formula to find the inverse of a matrix.
Marking: 4 marks for showing work, and 3 marks for the correct answer.
(d) Using the inverse of the coefficients matrix A^-1 that you calculated in the previous
File preview Ive for the (unknown) variables x, y, and z.
[6 marks]
Solution: Finally, we use the equation given in part (a) to find the solution:
Av = b
⇒ v = A^-1b = Cb
⇒ v = (1)/(58)
< p m a t r i x >

< p m a t r i x >
=
< p m a t r i x >
Marking: 3 marks for showing work, and 3 marks for the correct answer.

Added by Gavin R.

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