smc-617-30-Determinant

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, FUR1 2018 VCAA 1 MC

Which one of the following matrices has a determinant of zero?

A.	`[(0,1),(1,0)]`	B.	`[(1,0),(0,1)]`	C.	`[(1,2),(−3,6)]`

D.	`[(3,6),(2,4)]`	E.	`[(4,0),(0,−2)]`

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`D`

Show Worked Solution

`text(By trial and error:)`

`text(Consider option)\ D,`

`text(det)[(3,6),(2,4)]`	`= 3 xx 4 – 6 xx 2`
	`= 0`

`=> D`

MATRICES, FUR1 2017 VCAA 3 MC

Which one of the following matrix equations has a unique solution?

`[(1,1),(1,1)][(x),(y)] = [(2),(10)]`

`[(6,−6),(−4,4)][(x),(y)] = [(60),(36)]`

`[(8,−4),(4,2)][(x),(y)] = [(12),(18)]`

`[(7,0),(5,0)][(x),(y)] = [(14),(15)]`

`[(4,−2),(6,−3)][(x),(y)] = [(36),(24)]`

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`C`

Show Worked Solution

`text(Consider option)\ C:`

`Delta`	`= text(det)[(8,−4),(4,2)]`
	`= 8 xx 2 – (4 xx −4)`
	`= 32`
	`!= 0`

`text(In all other systems,)\ Delta = 0`

`=> C`

MATRICES, FUR1 2016 VCAA 3 MC

The matrix equation below represents a pair of simultaneous linear equations.

`[(12,9),(m,3)][(x),(y)] = [(6),(6)]`

These simultaneous linear equations have no unique solution when `m` is equal to

`−4`
`−3`
`0`
`3`
`4`

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`E`

Show Worked Solution

`[(x),(y)]`	`= [(12,9),(m,3)]^(−1)[(6),(6)]`
	`= 1/((12 xx 3) – (9 xx m)) [(3,−9),(−m,12)][(6),(6)]`

`text(No unique solution occurs when)\ \ Δ=0 :`

`(12 xx 3) – (9 xx m)`	`= 0`
`9m`	`= 36`
`m`	`= 4`

`=> E`

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) ---

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`[(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, FUR1 2008 VCAA 5 MC

The determinant of `[(3, 2), (6, x)]` is equal to 9.

The value of `x` is

A. `– 7`

B. `– 4.5`

C. `1`

D. `4.5`

E. `7`

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`E`

Show Worked Solution

`text(det) [(3, 2), (6, x)]`	`= 3x – 2 xx 6`
`:. 9`	`= 3x – 12`
`3x`	`= 21`
`x`	`= 7`

`=> E`

MATRICES, FUR1 2013 VCAA 4 MC

`2.8x + 0.7y`	`= 10`
`1.4x + ky`	`= 6`

The set of simultaneous linear equations above does not have a solution if `k` equals

A. `– 0.35`

B. `– 0.250`

C. `0`

D. `0.25`

E. `0.35`

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`E`

Show Worked Solution

`[(2.8,0.7),(1.4,k)][(x),(y)] = [(10),(6)]`

`text(det) [(2.8,0.7),(1.4,k)] = 2.8k – 0.7 xx 1.4`

`text(No solution if det) = 0,`

`0`	`= 2.8k – 0.98`
`k`	`= (0.98)/2.8`
	`= 0.35`

`rArr E`

MATRICES, FUR1 2006 VCAA 7 MC

How many of the following five sets of simultaneous linear equations have a unique solution?

A. 1

B. 2

C. 3

D. 4

E. 5

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`C`

Show Worked Solution

`text(Consider each set of equations,)`

`text(Set 1: det)[(4,2),(2,1)] = 0\ \ text{(not unique)}`

♦ Mean mark 42%.

`text(Set 2:)\ x = 0, x + y = 6\ \ text{(unique)}`

`text(Set 3: det)[(1,−1),(1,1)] = 2 != 0\ \ text{(unique)}`

`text(Set 4: det)[(2,1),(2,1)] = 0\ \ text{(not unique)}`

`text(Set 5:)\ x = 8, y = 2\ \ text{(unique)}`

`:. 3\ text(sets of equations have a unique solution.)`

`rArr C`

MATRICES, FUR1 2011 VCAA 3 MC

Each of the following four matrix equations represents a system of simultaneous linear equations.

`[(1,3),(0,2)] [(x),(y)]=[(4),(8)]`

`[(1,1),(2,2)] [(x),(y)]=[(5),(3)]`

`[(1,0),(0,2)] [(x),(y)]=[(4),(8)]`

`[(0,3),(0,2)] [(x),(y)]=[(6),(12)]`

How many of these systems of simultaneous linear equations have a unique solution?

A. 0

B. 1

C. 2

D. 3

E. 4

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`C`

Show Worked Solution

`text(Consider the 1st system,)`

`Delta = text(det)[(1,3),(0,2)] = 1 xx 2 – 3 xx 0 = 2 != 0`

`:.\ text(Unique solutions exists)`

`text(Similarly for the other systems, we find)`

`text(that)\ Delta != 0\ text{in two (total).}`

`=> C`

MATRICES, FUR1 2015 VCAA 3 MC

Four systems of simultaneous linear equations are shown below.

`12x + 8y`	`= 26`	`3x - 2y`	`= 14`	`−4x - 2y`	`= 17`	`x + 0.5y`	`= 8`
`3x + 2y`	`= 15`	`−7x + 5y`	`= 9`	`−6x + 3y`	`= 10`	`0.5x + y`	`= 8`

How many of these systems of simultaneous linear equations do not have a unique solution?

A. 0

B. 1

C. 2

D. 3

E. 4

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`B`

Show Worked Solution

`text(Looking at each system in turn,)`

`{:text(det):}[(12,8),(3,2)] = 12 xx 2 – 8 xx 3 = 0`

`{:text(det):}[(3,−2),(−7,5)] = 1 != 0`

`{:text(det):}[(−4,−2),(−6,3)] = −24 != 0`

`{:text(det):}[(1,0.5),(0.5,1)] != 0`

`:. 1\ text(system does not have a unique solution)`

`text{(i.e. det = 0).}`

`=> B`