smc-616-30-Matrix Product

Matrices, GEN1 2024 VCAA 25 MC

Matrix \(J\) is a \(2 \times 3\) matrix.

Matrix \(K\) is a \(3 \times 1\) matrix.

Matrix \(L\) is added to the product \(J K\).

The order of matrix \(L\) is

\(1 \times 3\)
\(2 \times 1\)
\(2 \times 3\)
\(3 \times 2\)

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\(B\)

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\(J\ \text{is order}\ 2\times 3\ \text{and}\ K\ \text{is order}\ \3\times 1\)

\(\therefore\ JK\ \text{is order}\ \2\times 1\)

\(\Rightarrow B\)

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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\(D\)

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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 26 MC

Matrix \(P\) is a permutation matrix and matrix \(Q\) is a column matrix.

\(P=\begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1
\end{bmatrix}
\quad \quad Q=\begin{bmatrix}
t \\
e \\
a \\
m \\
s
\end{bmatrix}\)

When \(Q\) is multiplied by \(P\), which three letters change position?

\(t, e, a\)
\(e, a, m\)
\(a, m, s\)
\(m, s, t\)
\(e, a, s\)

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\(B\)

Show Worked Solution

\(P \times Q =\begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
t \\
e \\
a \\
m \\
s
\end{bmatrix}
= \begin{bmatrix}
t \\
a \\
m \\
e \\
s
\end{bmatrix}\)

\(\Rightarrow B\)

MATRICES, FUR1 2020 VCAA 2 MC

Matrix `A = [(1, 2), (0, 3), (1, 0), (4, 5)]` and matrix `B = [(2, 0, 3, 1), (4, 5, 2, 0)]`.

Matrix `Q = A xx B`.

The element in row `i` and column `j` of matrix `Q` is `q_(ij)`.

Element `q_41` is determined by the calculation

`0 × 0 + 3 × 5`
`1 × 1 + 2 × 0`
`1 × 2 + 2 × 4`
`4 × 1 + 5 × 0`
`4 × 2 + 5 × 4`

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`E`

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`Q = [(1, 2), (0, 3), (1, 0), (4, 5)][(2, 0, 3, 1), (4, 5, 2, 0)]`

`q_41 = 4 xx 2 + 5 xx 4`

`=> 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 2018 VCAA 2 MC

The matrix product `[(4,2,0)] xx [(4),(12),(8)]` is equal to

`[144]`
`[(16),(24),(0)]`
`4 xx [(1,2,0)] xx [(1),(12),(8)]`
`2 xx [(2,1,0)] xx [(2),(6),(4)]`
`4 xx [(2,1,0)] xx [(2),(6),(4)]`

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`E`

Show Worked Solution

`[(4,2,0)] xx [(4),(12),(8)]`	`= 2[(2,1,0)] xx 2[(2),(6),(4)]`
	`= 4 xx [(2,1,0)] xx [(2),(6),(4)]`

`=> E`

MATRICES, FUR1 2016 VCAA 2 MC

The matrix product `[(0,0,0,1,0),(0,0,1,0,0),(1,0,0,0,0),(0,1,0,0,0),(0,0,0,0,1)] xx [(text(L)),(text(E)),(text(A)),(text(P)),(text(S))]` is equal to

`[(text(L)),(text(A)),(text(P)),(text(S)),(text(E))]`

`[(text(L)),(text(E)),(text(A)),(text(P)),(text(S))]`

`[(text(P)),(text(L)),(text(E)),(text(A)),(text(S))]`

`[(text(P)),(text(A)),(text(L)),(text(E)),(text(S))]`

`[(text(P)),(text(E)),(text(A)),(text(L)),(text(S))]`

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`D`

Show Worked Solution

`[(text(P)),(text(A)),(text(L)),(text(E)),(text(S))]`

`=> D`

MATRICES, FUR1 2010 VCAA 4 MC

Which matrix expression results in a matrix that contains the sum of the numbers 2, 5, 4, 1 and 8?

A. `[(1),(1),(1),(1),(1)] xx [(2, 5, 4, 1, 8)]`

B. `[(2, 5, 4, 1, 8)] xx [(1),(1),(1),(1),(1)]`

C. `[(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)] xx [(2, 0, 0, 0, 0), (0, 5, 0, 0, 0), (0, 0, 4, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 8)]`

D. `[(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)] xx [(2),(5),(4),(1),(8)]`

E. `[(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)] + [(2, 0, 0, 0, 0), (0, 5, 0, 0, 0), (0, 0, 4, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 8)]`

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`B`

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`text(The result needs to be a 1 × 1 matrix.)`

`=> B`

MATRICES, FUR1 2012 VCAA 9 MC

`[(3,4), (1,2)] × [(a),(3)] = [(6,3), (2,-1)] × [(2),(b)]`

Which set of equations below could be used to determine the values of `a` and `b` that are shown in the matrix equation above?

A. `a - b = 2`

`a + b = 0`

B. `a + b = -2`

`a - b = 0`

C. `a + b = 2`

`a - b = 0`

D. `a - b = 8`

`a + b = 2`

E. `a - b = 8`

`a + b = -2`

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`B`

Show Worked Solution

`text(Finding the matrix product of both sides:)`

`[(3,4),(1,2)][(a),(3)] = [(6,3),(2,−1)][(2),(b)]`

♦ Mean mark 39%.

`3a + 12`	`= 12 + 3b`
`a – b`	`= 0\ \ …\ (1)`

`a + 6`	`= 4 – b`
`a + b`	`=−2\ \ …\ (2)`

`rArr B`

MATRICES, FUR1 2012 VCAA 2 MC

If `A = [(8,1), (4,2)]` and `B= [(3,12),(6,0)],` then the matrix `AB = [(30,96), (24,48)].`

The element 24 in the matrix `AB` is correctly obtained by calculating

A. `4 × 6 + 2 × 0`

B. `4 × 3 + 2 × 6`

C. `3 × 4 + 12 × 1`

D. `4 × 2 + 8 × 2`

E. `8 × 3 + 1 × 0`

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`B`

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`[(8,1),(4,2)][(3,12),(6,0)] = [(30,96),(24,48)]`

`e_21\ text(in)\ AB\ text(is calculated.)`

`4 xx 3 + 2 xx 6 = 24`

`rArr B`

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 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`

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`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 2007 VCAA 3 MC

If `A = [(8, 4), (5, 3)]` and the product `AX = [(5, 6), (8, 10)]` then `X` is

A. `[(24, -14), (13, -7.5)]`	B. `[(-4.25, -5.5), (9.75, 12.5)]`

C. `[(-3.75, 7), (-6.5, 12)]`	D. `[(25, 11), (-19.5, -8.5)]`

E. `[(0.625, 1.5), (1.6, 3.333)]`

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`B`

Show Worked Solution

`AX`	`= [(5,6),(8,10)]`
`X`	`= [(8,4),(5,3)]^(−1)[(5,6),(8,10)]`
	`= [(0.75,−1),(−1.25,2)][(5,6),(8,10)]`
	`= [(−4.25,−5.5),(9.75,12.5)]`

`=> B`

MATRICES, FUR1 2011 VCAA 9 MC

Matrix `A` is a 3 x 3 matrix. Seven of the elements in matrix `A` are zero.

Matrix `B` contains six elements, none of which are zero.

Assuming the matrix product `AB` is defined, the minimum number of zero elements in the product matrix `AB` is

A. `0`

B. `1`

C. `2`

D. `4`

E. `6`

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`C`

Show Worked Solution

`text(Matrix)\ B\ text(is)\ 3 xx 2`

`text(If non-zero elements of matrix)\ A`

`text(are in the same row,)`

`[(a,0,b),(0,0,0),(0,0,0)][(c,d),(e,f),(g,h)] = [(x,y),(0,0),(0,0)]`

`text(where)\ a,b,c …, x, y != 0`

`text(If non-zero elements of matrix)\ B`

`text(are not in the same row,)`

`[(a,0,0),(b,0,0),(0,0,0)][(c,d),(e,f),(g,h)] = [(ac,ad),(bc,bd),(0,0)]`

`:.\ text(Minimum number of zero elements)`

`text(in matrix)\ AB\ text(is 2.)`

`=> C`