smc-617-20-SE to Matrix

Matrices, GEN1 2022 VCAA 6 MC

Consider the following system of simultaneous linear equations.

\(y+z=4\)

\(x-y+z=1\)

\(-x+y=2\)

The solution to these simultaneous equations can be found by calculating

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\(E\)

Show Worked Solution

\(
\begin{bmatrix}
0 & 1 & 1 \\ 1 & -1 & 1 \\
-1 & 1 & 0\end{bmatrix}
\times \begin{bmatrix}
x \\ y \\ z\end{bmatrix}
= \begin{bmatrix}
4 \\ 1 \\ 2\end{bmatrix}
\)

\(
\begin{bmatrix}
x \\ y \\ z\end{bmatrix}
= \begin{bmatrix}
0 & 1 & 1 \\
1 & -1 & 1 \\
-1 & 1 & 0
\end{bmatrix}^{-1}
\times \begin{bmatrix}
4 \\ 1 \\ 2
\end{bmatrix}
\)

\(
\begin{bmatrix}
x \\ y \\ z
\end{bmatrix}
= \begin{bmatrix}
1 & -1 & -2 \\
1 & -1 & -1 \\
0 & 1 & 1
\end{bmatrix}
\times\begin{bmatrix}
4 \\ 1 \\ 2\end{bmatrix}
\)

\(\Rightarrow E\)

♦♦ Mean mark 31%.

MATRICES, FUR1 2021 VCAA 3 MC

`ax + 4y = 10`

`18x + by = 6`

The set of simultaneous linear equations above does not have a unique solution when

`a = 2, \ b = 36`
`a = 3, \ b = 22`
`a = 4, \ b = 20`
`a = 5, \ b = 12`
`a = 6, \ b = 14`

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`A`

Show Worked Solution

`text{In matrix form:}`

`[(a,4),(18,b)] [(x),(y)] = [(10),(6)]`

`text{No unique solution} =>\ text{det} = 0`

`ab – 4 xx 18`	`= 0`
`ab`	`= 72`

`=> A`

MATRICES, FUR2 2006 VCAA 3

Market researchers claim that the ideal number of bookshops (`x`), sports shoe shops (`y`) and music stores (`z`) for a shopping centre can be determined by solving the equations

`2x + y + z = 12`

`x-y+z=1`

`2y-z=6`

Write the equations in matrix form using the following template. (1 mark)

`qquad[(qquadqquadqquadqquadqquad),(),()][(qquadquad),(qquadquad),(qquadquad)] = [(qquadquad),(qquadquad),(qquadquad)]`
Do the equations have a unique solution? Provide an explanation to justify your response. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
Write down an inverse matrix that can be used to solve these equations. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
Solve the equations and hence write down the estimated ideal number of bookshops, sports shoe shops and music stores for a shopping centre. (1 mark)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`[(2,1,1),(1,-1,1),(0,2,-1)][(x),(y),(z)] = [(12),(1),(6)]`
`text(Yes. See worked solutions.)`
`[(2,1,1),(1,-1,1),(0,2,-1)]^(-1) = [(-1,3,2),(1,-2,-1),(2,-4,-3)]`
`text(3 bookshops, 4 sports shoe shops, 2 music stores.)`

Show Worked Solution

a.	`[(2,1,1),(1,-1,1),(0,2,-1)][(x),(y),(z)] = [(12),(1),(6)]`

♦ Mean mark 35% for all parts (combined).

b.	`text(det)\ [(2,1,1),(1,-1,1),(0,2,-1)] = 1 != 0`

`:.\ text(A unique solution exists.)`

c. `text(By CAS,)`

`[(2,1,1),(1,-1,1),(0,2,-1)]^(-1) = [(-1,3,2),(1,-2,-1),(2,-4,-3)]`

d. `[(x),(y),(z)]= [(-1,3,2),(1,-2,-1),(2,-4,-3)][(12),(1),(6)]= [(3),(4),(2)]`

`:.\ text(Estimated ideal numbers are:)`

`text(3 bookshops)`

`text(4 shoe shops)`

`text(2 music stores)`

MATRICES, FUR2 2010 VCAA 3

The basketball coach has written three linear equations which can be used to predict the number of points, `p`, rebounds, `r`, and assists, `a`, that Oscar will have in his next game.

The equations are `p + r + a`	`= 33`
`2p - r + 3a`	`= 40`
`p + 2r + a`	`= 43`

These equations can be written equivalently in matrix form.
Complete the missing information below. (1 mark)

--- 0 WORK AREA LINES (style=lined) ---

`[(qquadqquad),(qquadqquad),(qquadqquad)][(p),(r),(a)] = [(33),(40),(43)]`

This matrix equation can be solved in the following way.

`[(p),(r),(a)] = [(7,-1,-4),(-1,0,1),(x,1,3)][(33),(40),(43)]`

Determine the value of `x` shown in the matrix equation above. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
How many rebounds is Oscar predicted to have in his next game? (1 mark)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

`[(1,1,1),(2,-1,3),(1,2,1)][(p),(r),(a)] = [(33),(40),(43)]`
`-5`
`10`

Show Worked Solution

a.	`[(1,1,1),(2,-1,3),(1,2,1)][(p),(r),(a)] = [(33),(40),(43)]`

b.	`[(7,-1,-4),(-1,0,1),(x,1,3)]`	`=[(1,1,1),(2,-1,3),(1,2,1)]^(-1)`
		`= [(7,-1,-4),(-1,0,1),(-5,1,3)]`

`:.x = -5`

c.	`[(p),(r),(a)]`	`= [(7,-1,-4),(-1,0,1),(-5,1,3)][(33),(40),(43)]`
		`= [(19),(10),(4)]`

`:.\ text(Oscar is predicted to have 10 rebounds in the next game.)`

MATRICES, FUR1 2012 VCAA 3 MC

`x + z`	`= 6`
`2y + z`	`= 8`
`2x + y + 2z`	`= 15`

The solution of the simultaneous equations above is given by

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`A`

Show Worked Solution

`[(1,0,1),(0,2,1),(2,1,2)][(x),(y),(z)] = [(6),(8),(15)]`

♦ Mean mark 41%.

`:. [(x),(y),(z)]`	`= [(1,0,1),(0,2,1),(2,1,2)]^(−1)[(6),(8),(15)]`
	`= [(−3,−1,2),(−2,0,1),(4,1,−2)][(6),(8),(15)]`

`rArr A`

MATRICES, FUR1 2007 VCAA 4 MC

Consider the following system of three simultaneous linear equations.

`2x+z=5`

`x-2y=0`

`y-z=-1`

This system of equations can be written in matrix form as

A. `[(2, 1), (1, -2), (1, -1)][(x), (y), (z)] = [(5), (0), (-1)]`	B. `[(2,0,1), (1,-2,0), (0,1, -1)][(x), (y), (z)] = [(5), (0), (-1)]`

C. `[(2, 1, 5), (1, -2, 0), (1, -1, -1)][(x), (y), (z)] = [(5), (0), (-1)]`	D. `[(2, 1, 0), (1, -2, 0), (1, -1, 0)][(x), (y), (z)] = [(5), (0), (-1)]`

E. `[(2, 1), (1, -2), (1, -1)][(5), (0), (-1)] = [(x), (y), (z)]`

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`B`

Show Worked Solution

`=> B`

MATRICES, FUR1 2014 VCAA 2 MC

`y - z`	`= 8`
`5x - y`	`= 0`
`x + z`	`= 4`

The system of three simultaneous linear equations above can be written in matrix form as

A.	`[[0,1,-1],[0,5,-1],[1,0,1]][[x],[y],[z]]=[[8],[0],[4]]`	B.	`[[0,1,-1],[5,-1,0],[1,0,1]][[x],[y],[z]]=[[8],[0],[4]]`

C.	`[[1,-1],[5,-1],[1,1]][[x],[y],[z]]=[[8],[0],[4]]`	D.	`[[0,5,1],[1,-1,0],[-1,0,1]][[x],[y],[z]]=[[8],[0],[4]]`

E.	`[[0,5,0],[-1,-1,0],[1,1,0]][[x],[y],[z]]=[[8],[0],[4]]`

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`B`

Show Worked Solution

`=>B`