smc-616-40-Powers/Inverse

Matrices, GEN1 2024 VCAA 27 MC

Consider the following matrix, where \(h \neq 0\).

\begin{bmatrix}
4 & g \\
8 & h
\end{bmatrix}

The inverse of this matrix does not exist when \(g\) is equal to

\(-2 h\)
\(\dfrac{h}{2}\)
\(h\)
\(2 h\)

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\(B\)

Show Worked Solution

\(\text{Inverse will not exist if }\ 4h-8g=0\)

\(\text{i.e. when}\ g=\dfrac{h}{2}\)

\(\Rightarrow B\)

Matrices, GEN1 2022 VCAA 5 MC

Matrix \(E\) is a 2 × 2 matrix.

Matrix \(F\) is a 2 × 3 matrix.

Matrix \(G\) is a 3 × 2 matrix.

Matrix \(H\) is a 3 × 3 matrix.

Which one of the following matrix products could have an inverse?

\(EF\)
\(FH\)
\(GE\)
\(GF\)
\(HG\)

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\(D\)

Show Worked Solution

Matrix product requires: [m × n] × [n × m]

Matrix inverse requires a square matrix: [m × n] × [m × n] = [m × m]

Option A: \(EF\) = [2 × 2] × [2 × 3] → [2 × 3] (not square, eliminate A)

Option B: \(FH\) = [2 × 3] × [3 × 3] → [2 × 3] (not square, eliminate B)

Option C: \(GE\) = [3 × 2] × [2 × 2] → [3 × 2] (not square, eliminate B)

Option D: \(GF\) = [3 × 2] × [2 × 3] → [3 × 3] (correct)

Option E: \(HG\) = [3 × 3] × [3 × 2] → [3 × 2] (not square, eliminate E)

\(\Rightarrow D\)

Matrices, GEN1 2023 VCAA 30 MC

How many of the following statements are true?

All square matrices have an inverse.
The inverse of a matrix could be the same as the transpose of that matrix.
If the determinant of a matrix is equal to zero, then the inverse does not exist.
It is possible to take the inverse of an identity matrix.

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\(D\)

Show Worked Solution

\(\text{All square matrices have an inverse → not true}\)

\(\text{The inverse of a matrix could be the same as the transpose of that matrix → true}\)

\(\text{If the determinant of a matrix is equal to zero, then the inverse does not exist → true}\)

\(\text{It is possible to take the inverse of an identity matrix → true}\)

\(\Rightarrow D\)

MATRICES, FUR2 2020 VCAA 2

The preferred number of cafes `(x)` and sandwich bars `(y)` in Grandmall’s food court can be determined by solving the following equations written in matrix form.

`[(5, -9),(4, -7)][(x),(y)]=[(7), (6)]`

The value of the determinant of the 2 × 2 matrix is 1.

Use this information to explain why this matrix has an inverse. (1 mark)
Write the three missing values of the inverse matrix that can be used to solve these equations. (1 mark)

`[(text( __), 9),(text( __), text( __)\ )]`

Determine the preferred number of sandwich bars for Grandmall’s food court. (1 mark)

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`text(S) text(ince determinant) = 1 != 0,`

`text(the matrix has an inverse)`
`[(-7, 9),(-4, 5)]`
`2`

Show Worked Solution

a. `text(S) text(ince determinant) = 1 != 0,`

♦ Mean mark part (a) 37%.

`->\ text(the matrix has an inverse)`

b. `[(-7, 9),(-4, 5)]`

Mean mark part (c) 51%.

c.	`[(x), (y)] = [(-7, 9), (-4, 5)][(7), (6)] = [(7),(2)]`

`:.\ text(Preferred number of sandwich bars) = 2`

MATRICES, FUR1 2020 VCAA 3 MC

Matrices `P` and `W` are defined below.

`P = [(0,0,1,0,0),(0,0,0,0,1),(0,1,0,0,0),(0,0,0,1,0),(1,0,0,0,0)] qquad qquad W = [(A),(S),(T),(O),(R)]`

If `P^n xx W = [(A),(S),(T),(O),(R)]`, the value of `n` could be

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`D`

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`text(Find)\ n\ text(where)\ P^n\ text(is a 5 × 5 identity matrix:)`

`P^4 = [(1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,0,0,0,1)]`

`=> D`

MATRICES, FUR1 2008 VCAA 4 MC

Matrix `A` is a `1 xx 3` matrix.

Matrix `B` is a `3 xx 1` matrix.

Which one of the following matrix expressions involving `A` and `B` is defined?

A. `A + 1/3 B`

B. `2B xx 3A`

C. `A^2 B`

D. `B^-1`

E. `B - A`

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`B`

Show Worked Solution

`text(Consider)\ B,`

`underset (3 xx 1) (2B) xx underset (1 xx 3) (3A)`

`text(S) text(ince the number of columns in)`

`text(matrix)\ 2B\ text(equals the number of)`

`text(rows in matrix)\ 3A,\ text(their product)`

`text(is defined.)`

`=> B`

MATRICES, FUR1 2015 VCAA 7 MC

Matrix `P` has inverse matrix `P^(−1)`.

Matrix `P` is multiplied by the scalar `w(w ≠ 0)` to form matrix `Q`.

Matrix `Q^(−1)` is equal to

A. `1/w P^(−1)`

B. `1/(w^2)P^(−1)`

C. `wP^(−1)`

D. `w^2P^(−1)`

E. `P^(−1)`

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`A`

Show Worked Solution

`Q = wP`

♦ Mean mark 43%.

`Q^(−1)`	`= (wP)^(−1)`
	`= 1/w P^(−1)`

`=> A`

MATRICES, FUR1 2006 VCAA 3 MC

Let `A = [(1,0), (0,1)], B = [(2,1), (1,0)]` and `C = [(1,-1), (-1,1)]`

Then `A^3 (B - C)` equals

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

C.	`[(3,6),(6,−3)]`	D.	`[(3,0),(0,−3)]`

E.	`[(5,10),(10,−5)]`

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`A`

Show Worked Solution

`A^3(B – C)`	`= [(1,0),(0,1)]^3([(2,1),(1,0)] – [(1,−1),(−1,1)])`
	`= [(1,0),(0,1)][(1,2),(2,−1)]`
	`= [(1,2),(2,−1)]`

`rArr A`

MATRICES, FUR1 2011 VCAA 8 MC

Considering the following matrix `A`.

`A = [(3,k),(-4,-3)]`

`A` is equal to its inverse `A^-1` for a particular value of `k`.

This value of `k` is

A. `– 4`

B. `– 2`

C. `0`

D. `2`

E. `4`

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`D`

Show Worked Solution

`A = A^(−1)`

♦ Mean mark 49%.

`[(3,k),(−4,−3)]`	`= 1/(−9 – (−4k)) [(−3,−k),(4,3)]`
`:. 3`	`= (−3)/(−9 + 4k)`
`1`	`=(-1)/(4k-9)`
`4k – 9`	`= −1`
`4k`	`= 8`
`k`	`= 2`

`=> D`

MATRICES, FUR1 2011 VCAA 4 MC

Matrix `A` is a 3 x 4 matrix.

Matrix `B` is a 3 x 3 matrix.

Which one of the following matrix expressions is defined?

A. `BA^2`

B. `BA - 2A`

C. `A + 2B`

D. `B^2 - AB`

E. `A^-1`

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`B`

Show Worked Solution

`text(Consider)\ B,`

`B`	`xx`	`A`	`=`	`BA`
`3 xx 3`		`3 xx 4`		`3 xx 4`

`:. BA`	`-`	`2Aqquadtext(is defined)`
`3 xx 4`		`3 xx 4`

`text(All other options can be shown)`

`text(to produce undefined matrices.)`

`=> B`