eJournal 6: Matrices and Transformations

Adding and subtracting Matrices:

Matrices can only be added or subtracted if they have the same order of matrices. Here is an example where two matrices could not be added together. The same applies for subtracting matrices.

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Multiplying Matrices:

Scalar Multiplication – the constant is referred to as a scalar.

Matrices can only be multiplied if the ‘middle of the order of matrices’ are the same. For example, the above matrices have the order of (2×3) and (3×2) respectively, and the middle number 3 is the same, so these two matrices can be multiplied together.

Dividing matrices:

Actually, matrices cannot be divided. However, lets say matrix A can be obtained from matrix AB by multiplying AB with the inverse of A.

A/B = A × (1/B) = A × B^-1

where B^-1 means the “inverse” of B.

So, how is the inverse of a matrix obtained?

A × A^-1 = I

where I is the identity matrix

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Identity matrices can only be ‘square’ matrices, such as 2×2, 3×3, 4×4, etc.

Transformations

There are 4 types of matrix transformations:

• Translations
• Reflections
• Enlargements
• Rotations

Translations

To “slide” an image without rotating or flipping it. A translated image has the same size and shape of the original image.

Reflections

Reflected images

• Have the exact the same size as its original object.
• Are laterally inverted compared to the original object.
• Have the same perpendicular distance to the mirror as compared to the object.
• IGNORE THIS (“Image is virtual – cannot be captured on a screen.”)

Enlargements

Nothing much… just making an image bigger or smaller given the centre of enlargement.

If the the scalar of enlargement is negative, for example -2, then the image would be the object rotated 180 degrees but 2 times smaller. If the the scalar of enlargement is a negative fraction, for example -1/3, then the image would be the object rotated 180 degrees but 3 times bigger.

Rotations

Turning around a centre of rotation. The distance from the centre of rotation to any side of the shape will always remain constant.

Real Life Applications of Matrices:

After all this tedious working, you might be wondering how on earth these might be applied in real life!? Here’s how:

EXAMPLE 1

Now think about this … the value of sales for Monday is calculated this way:Apple pie value + Cherry pie value + Blueberry pie value$3×13 + $4×8 + $2×6 = $83

So it is, in fact, the “dot product” of prices and how many were sold:

($3, $4, $2) • (13, 8, 6) = $3×13 + $4×8 + $2×6  
    = $83

We match the price to how many sold, multiply each, then sum the result.

In other words:

• The sales for Monday were: Apple pies: $3×13=$39, Cherry pies: $4×8=$32, and Blueberry pies: $2×6=$12. Together that is $39 + $32 + $12 = $83
• And for Tueday: $3×9 + $4×7 + $2×4 = $63
• And for Wednesday: $3×7 + $4×4 + $2×0 = $37
• And for Thursday: $3×15 + $4×6 + $2×3 = $75

So it is important to match each price to each quantity.

EXAMPLE 2

EXAMPLE 3

In geology, matrices are used in taking seismic surveys. They are used for plotting graphs, statistics and also to do scientific studies in almost different fields.

While completing this blog, I learnt the IB value of being open-minded as the real life applications of matrices were an eye opener to me, as I realise that many facts are yet to be discovered (outside the textbook). I also learnt the IB value of being a thinker as I realised my mind has to be flexible in remembering the different methods required such as multiplying matrices together.