Matrix Multiplication

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Name	Status	Author	Last Modified
Matrix Multiplication	Ready to use	Mark Patterson	20/09/2020 20:29
Alexander's copy of Matrix Multiplication	Ready to use	Alexander Corner	06/09/2021 15:41
Alexander's copy of Matrix Multiplication 2	Ready to use	Alexander Corner	06/09/2021 15:42

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Question statement

Answer the following questions on matrix multiplication.

Give any introductory information the student needs.

			Name	Type	Generated Value

maS1	matrix	matrix([8,-2],[6,7])
rand1	integer	-2
maS2	matrix	matrix([8,9,0],[3,2,6])
rand2	integer	7
maS3	matrix	matrix([2,3,3,9])
rand3	integer	6
maMa	matrix	matrix([8,6],[7,8],[1,8])
maMb	matrix	Matrix of size 3×3
maMc	matrix	matrix([6,1,9,3],[1,2,5,5])
maMd	matrix	Matrix of size 4×3
maMe	matrix	Matrix of size 4×2
maS1a	matrix	matrix([-16,4],[-12,-14])
maMac	matrix	Matrix of size 3×4

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
matrixrandom(2,2)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `matrix`

matrix([8,-2],[6,7])

→ Used by:

maS1a

Advice
"Part a)" - prompt
"Part a)" → "Gap 0." - Correct answer

Parts

Information only
Part a) Gap-fill
1. Gap Gap 0. Matrix entry
  Add
2. Gap Gap 1. Matrix entry
  Add
3. Gap Gap 2. Matrix entry
  Add
Information only
Part b) Match choices with answers
Part c) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
  Add
4. Gap Gap 3. Number entry
  Add
5. Gap Gap 4. Number entry
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6. Gap Gap 5. Number entry
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7. Gap Gap 6. Number entry
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8. Gap Gap 7. Number entry
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Part d) Matrix entry
Part e) Matrix entry
Part f) Matrix entry

Information only

Ask the student a question, and give any hints about how they should answer this part.

Scalar Multiplication

Advice

Scalar Multiplication

For scalar multiplication, multiply each element by the scalar value. The resultant matrix will be of the same order (size) as the starting matrix.

e.g. for the first question:

$\simplify{{maS1}}\times\var{rand1} = \begin{pmatrix} \var{maS1[0][0]}\times\var{rand1} & \var{maS1[0][1]}\times\var{rand1} \\ \var{maS1[1][0]}\times\var{rand1} & \var{maS1[1][1]}\times\var{rand1} \end{pmatrix} = \simplify{{maS1a}}$

Matrix by Matrix Multiplcation

Matrices can only be multiplied together when the number of columns in the first matrix is equal to the number of rows in the second matrix. The resultant matrix will have the same number of rows as the first matrix, and the same number of columns as the second matrix.

For $AC$ ,

$A = \simplify{{maMa}} C = \simplify{{maMc}}$

The order of $A$ is [3 x 2] and the order of $C$ is [2 x 4].

[3 x 2] x [2 x 4] : The inner numbers match, therefore these matrices can be multiplied. The outer numbers give the size of the new matrix, which will be [3 x 4].

The answer matrix will look like this: $\begin{pmatrix} ac_{11} & ac_{12} & ac_{13} & ac_{14} \\ ac_{21} & ac_{22} & ac_{23} & ac_{24} \\ ac_{31} & ac_{32} & ac_{33} & ac_{34} \end{pmatrix}$

To work out the values for the answer matrix we multiply:-

Each value in row 1 in matrix $A$ by the corresponding value in column 1 in matrix $C$ – and work out the total. This provides the value for $ac_{11}$
Each value in row 1 in matrix $A$ by the corresponding value in column 2 in matrix $C$ – and work out the total. This provides the value for $ac_{12}$
Each value in row 1 in matrix $A$ by the corresponding value in column 3 in matrix $C$ – and work out the total. This provides the value for $ac_{13}$
Each value in row 1 in matrix $A$ by the corresponding value in column 4 in matrix $C$ – and work out the total. This provides the value for $ac_{14}$
Each value in row 2 in matrix $A$ by the corresponding value in column 1 in matrix $C$ – and work out the total. This provides the value for $ac_{21}$
and so on for each corresponding element in the answer matrix.

So:

$ac_{11} = a_{11}c_{11}+a_{12}c_{21} = \var{maMa[0][0]}\times\var{maMc[0][0]} + \var{maMa[0][1]}\times\var{maMc[1][0]} = \var{maMac[0][0]}$

$ac_{12} = a_{11}c_{12}+a_{12}c_{22} = \var{maMa[0][0]}\times\var{maMc[0][1]} + \var{maMa[0][1]}\times\var{maMc[1][1]} = \var{maMac[0][1]}$

$ac_{13} = a_{11}c_{13}+a_{12}c_{23} = \var{maMa[0][0]}\times\var{maMc[0][2]} + \var{maMa[0][1]}\times\var{maMc[1][2]} = \var{maMac[0][2]}$

$ac_{14} = a_{11}c_{14}+a_{12}c_{24} = \var{maMa[0][0]}\times\var{maMc[0][3]} + \var{maMa[0][1]}\times\var{maMc[1][3]} = \var{maMac[0][3]}$

$ac_{21} = a_{21}c_{11}+a_{22}c_{21} = \var{maMa[1][0]}\times\var{maMc[0][0]} + \var{maMa[1][1]}\times\var{maMc[1][0]} = \var{maMac[1][0]}$

and so on...

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Matrix Multiplication

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Mark Patterson 4 years, 6 months ago

Mark Patterson 4 years, 6 months ago

Mark Patterson 4 years, 6 months ago