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[(qquadqquadqquad),(),()][(quad),(quad),(quad)] = [(quad),(quad),(quad)]`
Do the equations have a unique solution? Provide an explanation to justify your response. (1 mark)
Write down an inverse matrix that can be used to solve these equations. (1 mark)
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)

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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`