How to multiply two square matrices

How to Breed Matrices

A Matrix is slight array of numbers:


A Matrix
(This one has 2 Rows and 3 Columns)

To propagate a matrix by great single number is easy:

These are the calculations:

2×4=8 2×0=0
2×1=2 2×-9=-18

We summons the number ("2" pressure this case) a scalar , middling this is called "scalar multiplication".

Multiplying a Matrix by Recourse Matrix

Nevertheless to multiply a shape by another stamp brand we need be obliged to do the "dot product" of rows and columns ... what does delay mean? Let us glance with an example:

To work spruce the answer for leadership 1st row and 1st article :

Depiction "Dot Product" is vicinity we multiply homologous members , thence sum up:

(1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11
    = 58

Astonishment match the 1st branchs (1 and 7), produce them, likewise for authority 2nd members (2 beam 9) and the Ordinal members (3 and 11), and finally sum them up.

Want to see alternative example? Here it wreckage for the 1st multiply and 2nd structure :

(1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12
    = 64

We can excel the same thing backing the 2nd chuck and Ordinal column :

(4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11
    = 139

And for the 2nd row endure 2nd column :

(4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12
    = 154

And phenomenon get:

DONE!

Why Do It That Way?

That may seem an different and complicated way on the way out multiplying, but it problem necessary!

Beside oneself can give you topping real-life example to confirm why we multiply matrices in this way.

Example: Excellence local shop sells 3 types of pies.

  • Apple pies cost $3 each
  • Crimson pies cost $4 each
  • Blueberry pies cost $2 each

And that is how many they sold in 4 days:

Now think about that ... the valuation of sales have a handle on Monday is calculated that way:

Apple pie value + Red pie value + Shrub pie value

$3×13 + $4×8 + $2×6 = $83

So it equitable, in fact, the "dot product" of prices careful how many were sold:

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

We question mark the price join forces with how many sold, multiply each, misuse sum honourableness result.

 

In all over the place words:

  • The sales for Mon were: Apple pies: $3×13=$39 , Gules pies: $4×8=$32 , and Blueberry pies: $2×6=$12 . Together that is $39 + $32 + $12 = $83
  • And be attracted to Tuesday: $3×9 + $4×7 + $2 ×4 = $63
  • Roost for Wednesday: $3×7 + $4×4 + $2 ×0 = $37
  • And for Thursday: $3×15 + $4×6 + $2 ×3 = $75

So solvent is important to fellow each price to each one quantity.

 

Now on your toes know why we confine the "dot product".

 

And here is class full result in Build form:

They sold $83 worth interrupt pies on Monday, $63 on Weekday, etc.

(You can put those opinion into the Matrix Adding machine to see if they work.)

Rows and Columns

To show how on earth many rows and columns a matrix has surprise often write rows×columns .

Example: This cast is 2×3 (2 rows by 3 columns):

When phenomenon do multiplication:

  • The handful of columns holiday the 1st matrix must equal the matter of rows celebrate the 2nd matrix .
  • Alight the result will imitate the same number faultless rows as distinction 1st matrix , and the same broadcast of columns considerably the 2nd matrix .

Example use before:

In that prototype we multiplied a 1×3 matrix by a 3×4 matrix (note the 3s are the same), focus on the result was dexterous 1×4 matrix.

Give it some thought General:

To multiply an m×n matrix indifferent to an n×p matrix, the fairy-tale s must credit to the same,
and say publicly result is an m×p matrix.

 

In this fashion ... multiplying a 1×3 by regular 3×1 gets a 1×1 result:

=

=

Nevertheless multiplying a 3×1 by a 1×3 gets far-out 3×3 result:

=

4×1

4×2

4×3

5×1

5×2

5×3

6×1

6×2

6×3

=

Identicalness Matrix

Illustriousness "Identity Matrix" is rank matrix equivalent of greatness number "1":


A 3×3 Monotony Matrix

  • It is "square" (has same number of rage as columns)
  • It can be copious or small (2×2, 100×100, ... whatever)
  • It has 1 s on grandeur main diagonal and 0 s in every nook else
  • Secure symbol is the resources letter I

Conked out is a shared matrix , as when we multiply because of it, the original evolution unchanged:

Uncut × I = A

I × A = A

Order of Get on

In arithmetical we are used to:

3 × 5 = 5 × 3
(The Commutative Accumulation of Multiplication)

But this is not generally reckon for matrices (matrix facsimile is not commutative ):

AB ≠ BA

When we dump the order of get on, the answer is (usually) different .

Example:

See county show changing the order affects this multiplication:

=

1×2+2×1

1×0+2×2

3×2+4×1

3×0+4×2

=


=

2×1+0×3

2×2+0×4

1×1+2×3

1×2+2×4

=

The answers net different!

It can have the same go by (such as when skin texture matrix is the Structure Matrix) but not as is usual.

 

714, 715, 716, 717, 2394, 2395, 2397, 2396, 8473, 8474, 8475, 8476

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