smc-616-40-Powers/Inverse

Matrices, GEN1 2024 NHT 27 MC

Matrix \(V\) is an \(n \times n\) matrix with a determinant equal to 1 .

The product of \(V \times V^{-1}\) will result in

an identity matrix.
a Leslie matrix.
a column matrix.
a zero matrix.
a row matrix.

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\(A\)

Show Worked Solution

\(\text{Since matrix determinant}\ \neq 0\ \ \Rightarrow\ \ \text{matrix has an inverse}\)

\(\therefore V \times V^{-1} = I\)

\(\Rightarrow A\)

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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\(B\)

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\(\text{Inverse will not exist if determinant = 0}\)

\(\text{Find}\ g\ \text{when}\ \ 4h-8g=0:\)

\(g=\dfrac{h}{2}\)

\(\Rightarrow B\)

♦ Mean mark 50%.

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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\(D\)

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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)

--- 2 WORK AREA LINES (style=lined) ---
Write the three missing values of the inverse matrix that can be used to solve these equations. (1 mark)

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

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

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

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

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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`