Hello Everyone,This was a bit of an odd week with many people in the school falling to a illness and the postponing of the open house.We worked on many of the same projects and continued with studying Japan and its world view.We have arrived at the time in the narritive when Japan must choose how it will go forward in the face of being forced to interact with the west. I also have a longer list of multiplication tips and tricks if you are interested. You can test yourself using the times tables test. You can try out Speed Math and Fix The Equation. Learn Long Multiplication. And if you are really good, see if you can beat the high scores at Reaction Math.
A Matrix is an array of numbers:
This multiplication chart is appropriate for Kindergarten, 1st Grade, 2nd Grade, 3rd Grade, 4th Grade, and 5th Grade. Multiplication Times Tables Practice Worksheets These multiplication times table practice worksheets may be used with four different times table ranges starting at 1 through 9 and going up to 1 through 12. Jul 02, 2020 To do long multiplication quickly, start by splitting up the tens and ones place in the smaller number. For example, if the number was 12, you would end up with 10 and 2. Next, multiply the bigger number by both the tens number and the ones number. This Week We Will Focus On: Reading Comprehension Focus.
A Matrix
(This one has 2 Rows and 3 Columns)
To multiply a matrix by a single number is easy:
| 2×4=8 | 2×0=0 |
| 2×1=2 | 2×-9=-18 |
We call the number ('2' in this case) a scalar, so this is called 'scalar multiplication'.
Multiplying a Matrix by Another Matrix
But to multiply a matrix by another matrix we need to do the 'dot product' of rows and columns ... what does that mean? Let us see with an example:
To work out the answer for the 1st row and 1st column:
The 'Dot Product' is where we multiply matching members, then sum up:
(1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11
= 58
We match the 1st members (1 and 7), multiply them, likewise for the 2nd members (2 and 9) and the 3rd members (3 and 11), and finally sum them up.
Want to see another example? Here it is for the 1st row and 2nd column:
(1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12
= 64
We can do the same thing for the 2nd row and 1st column:
(4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11
= 139
And for the 2nd row and 2nd column:
(4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12
= 154
And we get:
DONE!
Why Do It This Way?
This may seem an odd and complicated way of multiplying, but it is necessary!
I can give you a real-life example to illustrate why we multiply matrices in this way.
Example: The local shop sells 3 types of pies.
- Apple pies cost $3 each
- Cherry pies cost $4 each
- Blueberry pies cost $2 each
Multiplication Chart
And this is how many they sold in 4 days:
Now think about this ... the value of sales for Monday is calculated this way:
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 Tuesday: $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.
Now you know why we use the 'dot product'.
And here is the full result in Matrix form:
They sold $83 worth of pies on Monday, $63 on Tuesday, etc.
(You can put those values into the Matrix Calculator to see if they work.)
Rows and Columns
To show how many rows and columns a matrix has we often write rows×columns.
Example: This matrix is 2×3 (2 rows by 3 columns):
When we do multiplication:
- The number of columns of the 1st matrix must equal the number of rows of the 2nd matrix.
- And the result will have the same number of rows as the 1st matrix, and the same number of columns as the 2nd matrix.
Example:
In that example we multiplied a 1×3 matrix by a 3×4 matrix (note the 3s are the same), and the result was a 1×4 matrix.
In General:
To multiply an m×n matrix by an n×p matrix, the ns must be the same,
and the result is an m×p matrix.
So ... multiplying a 1×3 by a 3×1 gets a 1×1 result:
But multiplying a 3×1 by a 1×3 gets a 3×3 result:
Multiplication Table
Identity Matrix
The 'Identity Matrix' is the matrix equivalent of the number '1':
A 3×3 Identity Matrix
- It is 'square' (has same number of rows as columns)
- It can be large or small (2×2, 100×100, ... whatever)
- It has 1s on the main diagonal and 0s everywhere else
- Its symbol is the capital letter I
It is a special matrix, because when we multiply by it, the original is unchanged:
A × I = A
I × A = A
Order of Multiplication
In arithmetic we are used to:
3 × 5 = 5 × 3
(The Commutative Law of Multiplication)
But this is not generally true for matrices (matrix multiplication is not commutative):
AB ≠ BA
When we change the order of multiplication, the answer is (usually) different.
Example:
See how changing the order affects this multiplication:
Multiplication Definition
Multiplication Games
The answers are different!
It can have the same result (such as when one matrix is the Identity Matrix) but not usually.