smc-616-20-Order / (Un)Defined

MATRICES, FUR2 2021 VCAA 1

Elena imports three brands of olive oil: Carmani (`C`) Linelli (`L`) and Ohana (`O`).

The number of 1 litre bottles of these oils sold in January 2021 is shown in matrix `J` below.

`J = {:[(2800),(1700),(2400)]:} {:(C),(L),(O):}`

What is the order of matrix `J`? (1 mark)
Elena expected that in February 2021 the sales of all three brands of olive oil would increase by 5%.
She multiplied matrix `J` by a scalar value, `k` , to determine the expected volume of sales for February.
Write down the value of the scalar `k`. (1 mark)

`k =`

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`3 xx 1`
`1.05`

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a. `3 xx 1`

b. `text{All brands} \ ↑ 5%`

`:. k = 1.05`

MATRICES, FUR1 2021 VCAA 5 MC

`A` is a 7 × 7 matrix.

`B` is a 10 × 7 matrix.

Which one of the following matrix expressions is defined?

`AB - 2B`
`A(BA)^-1`
`AB^2`
`A^2 - BA`
`A(B^T)`

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`E`

Show Worked Solution

`text{Consider each option}`

♦ Mean mark 45%.

`A : \ AB-2B -> text{not defined (AB not defined)}`

`B : \ A(BA)^-1`

`BA -> (10 xx 7) (7 xx 7) -> (10 xx 7)`

`( BA)^-1 \ text{is not defined (inverse matrices only apply to square matrices)}`

`C : \ AB^2 -> text{not defined} \ (B^2 \ text{not defined – only square matrices can be raised to a power)}`

`D : \ A^2 – BA`

`A^2 -> (7 xx 7), \ BA -> (10 xx 7)`

`qquad \ A^2 – BA \ -> text{not defined (only matrices of the same order can be subtracted)}`

`E: \ A(B^T)`

`A -> (7 xx 7), \ B^T -> (7 xx 10)`

`A (B^T) \ text{is defined as Columns in} \ A = \ text{rows in} \ B^T`

`=> E`

MATRICES, FUR1-NHT 2019 VCAA 6 MC

Matrix `W` is a `3 xx 2` matrix.

Matrix `Q` is a matrix such that `Q xx W = W`.

Matrix `Q` could be

A.	`\ [1]`	B.	`\ [(1,0),(0,1)]`	C.	`\ [(1,1),(1,1),(1,1)]`
D.	`\ [(1,0,0),(0,1,0),(0,0,1)]`	E.	`\ [(1,1,1),(1,1,1),(1,1,1)]`

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`D`

Show Worked Solution

`W = (3 xx 2)`

`[(1,0,0),(0,1,0),(0,0,1)][(e_11,e_12),(e_21,e_22),(e_31,e_32)] = [(e_11,e_12),(e_21,e_22),(e_31,e_32)]`

`=>\ D`

MATRICES, FUR1 2019 VCAA 1 MC

Consider the following four matrix expressions.

`[(8), (12)]+[(4), (2)] qquad qquad qquad qquad qquad qquad quad [(8), (12)]+[(4, 0),(0, 2)]`

`[(8, 0),(12, 0)] + [(4), (2)] qquad qquad qquad qquad qquad \ [(8, 0),(12, 0)] + [(4, 0),(0, 2)]`

How many of these four matrix expressions are defined?

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`C`

Show Worked Solution

`[(8), (12)]+[(4), (2)] = [(12), (4)]`

`[(8), (12)]+[(4, 0),(0, 2)] = text(not defined)`

`[(8, 0),(12, 0)] + [(4), (2)] = text(not defined)`

`[(8, 0),(12, 0)] + [(4, 0),(0, 2)] = [(12, 0), (12, 2)]`

`=> C`

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 2010 VCAA 8 MC

`m` and `n` are positive whole numbers.

Matrix `P` is of order `m xx n.`

Matrix `Q` is of order `n xx m.`

The matrix products `PQ` and `QP` are both defined

A. for no values of `m` and `n`

B. when `m` is equal to `n` only

C. when `m` is greater than `n` only

D. when `m` is less than `n` only

E. for all values of `m` and `n.`

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`E`

Show Worked Solution

`underset (m xx n) P xx underset (n xx m) Q = underset (m xx m)(PQ)`

`:. PQ\ text(is defined for all values)`

`text(of)\ m and n.`

`text(Similarly,)`

`underset (n xx m)Q xx underset (m xx n) P = underset (n xx n) (QP)`

`=> E`

MATRICES, FUR1 2010 VCAA 1 MC

The order of the matrix `[(2, 2), (2, 2), (2, 2)]` is

A. `2 xx 2`

B. `2 xx 3`

C. `3 xx 2`

D. `4`

E. `6`

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`C`

Show Worked Solution

`=> C`

MATRICES, FUR1 2013 VCAA 9 MC

`P, Q, R` and `S` are matrices such that the matrix product `P = QRS` is defined.

Matrix `Q` and matrix `S` are square, non-zero matrices for which `Q + S` is not defined.

Which one of the following matrix expressions is defined?

A. `R - S`

B. `Q + R`

C. `P^2`

D. `R^-1`

E. `P × S`

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`E`

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`text(If)\ Q(n xx n)\ text(and)\ S(m xx m)\ text(are square)`

♦ Mean mark 39%.

`text(matrices and)\ Q + S\ text(is not defined,)`

`m != n.`

`text(S)text(ince)\ QRS\ text(is defined, we can deduce)`

`text(that)\ R\ text(is)\ n xx m,\ text(and)\ P\ text(is)\ n xx m.`

`text(Consider)\ E,`

`P xx S\ text(is defined because the number)`

`text(of columns in)\ P(m)\ text(equals the number)`

`text(of rows in)\ S(m).`

`text(All other options can be shown to not)`

`text(be defined.)`

`rArr E`

MATRICES, FUR1 2013 VCAA 2 MC

Matrix `A` has three rows and two columns.

Matrix `B` has four rows and three columns.

Matrix `C = B × A` has

A. two rows and three columns.

B. three rows and two columns.

C. three rows and three columns.

D. four rows and two columns.

E. four rows and three columns.

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`D`

Show Worked Solution

`B`	`xx`	`A`	`=`	`C`
`4 xx 3`		`3 xx 2`		`4 xx 2`

`rArr D`

MATRICES, FUR1 2006 VCAA 6 MC

If `A = [(1,3), (6,4), (0,0)]` and the matrix product `XA = [(4,1),(1,4), (3,5)],` then the order matrix `X` is

A. `(2 xx 2)`

B. `(2 xx 3)`

C. `(3 xx 1)`

D. `(3 xx 2)`

E. `(3 xx 3)`

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`E`

Show Worked Solution

`X`	`xx`	`A`	`=`	`XA`
`a xx b`		`3 xx 2`		`3 xx 2`

♦♦♦ Mean mark 23%.

`b = 3\ \ text{(columns in}\ X = text(rows in)\ A)`

`a = 3\ \ text{(rows in}\ X = text(rows in)\ XA)`

`:. XA\ text(is order)\ 3 xx 3.`

`rArr E`

MATRICES, FUR1 2006 VCAA 2 MC

Let `A = [(-2), (0)], B = [(0, 9)]` and `C = [2]`

Using these matrices, the matrix product that is not defined is

A. `AB`

B. `AC`

C. `BA`

D. `BC`

E. `CB`

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`D`

Show Worked Solution

`text(Consider)\ D,`

`B`	`xx`	`C`
`1 xx 2`		`1 xx 1`

`text(Columns in)\ B !=\ text(Rows in)\ C`

`:. BC\ text(is undefined.)`

`rArr D`

MATRICES, FUR1 2007 VCAA 9 MC

Matrix `M` is a `3 xx 4` matrix.

Matrix `P` has five rows.

`N` is another matrix.

If the matrix product

`M (NP) = [(4, 1, 7, 2), (0, 9, 7, 4), (4, 3, 3, 1)],`

then the order of matrix `N` is

A. `3 xx 5`

B. `5 xx 3`

C. `4 xx 5`

D. `5 xx 4`

E. `5 xx 5`

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`C`

Show Worked Solution

`M xx (NP) = [(4,1,7,2),(0,9,7,4),(4,3,3,1)]`

♦ Mean mark 50%.

`text{The order of matrices (above) is}`

`(3 xx 4) xx ((a xx b)(5 xx c)) = (3 xx 4)`

`a = 4, b = 5, c = 4`

`:. N\ text{is a (4 × 5) matrix.}`

`=> C`

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`