Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history. X(x+1) is WRONG! It means a function x of x-1. X.(x-1) is right. Use the ^ (caret) for exponentiation. X^2 means x squared. X/2y means Resources: Simplifier Portal, help with entering simplifier formulas (a must read). X,Y = meshgrid(x,y) returns 2-D grid coordinates based on the coordinates contained in vectors x and y. X is a matrix where each row is a copy of x, and Y is a matrix where each column is a copy of y. The bases are 2 and 2 and the logarithmic numbers are x and x/4. I'm pulling my hair out. I know I could break it down to something like this: (log2x)(log2x) + (log2x)(log2x) -(log2 x/4) -(log2 x/4) -(log2 x/4) -11 = 0 After that I'm probably supposed to use log(x/y)=logx-logy, but I'm not sure how, and after that I'm not sure what to do.:-/ I know x is somewhere between 5.
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Matrix Multiplication / The Identity Matrix (page 3 of 3)

Here are a couple more examples of matrix multiplication:

  • Find CD and DC, if they exist, given that C and D are the following matrices:
    • C is a 3×2 matrix and D is a 2×4 matrix, so first I'll look at the dimension product for CD:
    So the product CD is defined (that is, I can do the multiplication); also, I can tell that I'm going to get a 3×4 matrix for my answer. Here's the multiplication:
    However, look at the dimension product for DC:
    Since the inner dimensions don't match, I can't do the multiplication. (The columns of C aren't the same length as the rows of D; the columns of C are too short, or, if you prefer, the rows of D are too long.) Then the answer is:
      DC is not defined.
  • Given the following matrices A and B, and defining C as AB = C, find the values of entries c3,2 and c2,3 in matrix C.
The dimension product of AB is (4×4)(4×3), so the multiplication will work, and C will be a 4×3 matrix. But to find c3,2, I don't need to do the whole matrix multiplication. The 3,2-entry is the result of multiplying the third row of A against the second column of B, so I'll just do that:
c3,2 = (3)(3) + (–2)(4) + (–2)(0) + (–2)(–1) = 9 – 8 + 0 + 2 = 3
On the other hand, c2,3 is the result of multiplying the second row of A against the third column of B, so: Copyright © Elizabeth Stapel 2003-2011 All Rights Reserved
c2,3 = (0)(0) + (2)(–2) + (1)(–2) + (4)(0) = 0 – 4 – 2 + 0 = –6
c3,2 = 3 and c2,3= –6
This type of problem serves as a reminder that, in general, to find ci,jyou multiply row i of A against column j of B.

Multiplying by the identity

The 'identity' matrix is a square matrix with 1's on the diagonal and zeroes everywhere else. Multiplying a matrix by the identity matrix I (that's the capital letter 'eye') doesn't change anything, just like multiplying a number by 1 doesn't change anything. This property (of leaving things unchanged by multiplication) is why I and 1 are each called the 'multiplicative identity' (the first for matrix multiplication, the latter for numerical multiplication).
But while there is only one 'multiplicative identity' for regular numbers (being the number 1), there are lots of different identity matrices. Why? Because the identity matrix you need for any particular matrix multiplication will depend upon the size of the matrix against which the identity is being multiplied, and perhaps also the side against which you're doing the multiplication (because, for a non-square matrix, right-multiplication and left-multiplication will require a different-size identity matrix).
Motrix
For instance, suppose you have the following matrix A:
To multiply A on the right by the identity (that is, to do AI ), you have to use I3, the 3×3 identity, in order to have the right number of rows for the multiplication to work:
On the other hand, to multiply A on the left by the identity, you have to use I2, the 2×2 identity, in order to have the right number of columns:
That is, if you are dealing with a non-square matrix (such as A in the above example), the identity matrix you use will depend upon the side that you're multiplying on. This is just another example of matrix weirdness. Don't let it scare you. Matrices aren't bad; they're just different.. really, really different.

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Cite this article as:
Stapel, Elizabeth. 'Matrix Multiplication / The Identity Matrix.' Purplemath. Available from
https://www.purplemath.com/modules/mtrxmult3.htm. Accessed [Date] [Month] 2016
 Faststone image viewer mac.
A Matrix is an array of numbers:


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

To multiply a matrix by a single number is easy:
2×4=82×0=0
2×1=22×-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:
X 2 1 x 2 4
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

X 2 1 X 2 4

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.

And this is how many they sold in 4 days:
Now think about this .. the value of sales for Monday is calculated this way:
Apple pie value + Cherry pie value + Blueberry pie value
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:
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:

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.

Handy note 1 0 9. 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:
2
5
=
=
But multiplying a 3×1 by a 1×3 gets a 3×3 result:
5
2
=
4×2
5×1
5×3
6×2
=
8
5
15
12

Identity Matrix

The 'Identity Matrix' is the matrix equivalent of the number '1':


A 3×3 Identity Matrix

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

Motrix 1 1 3 X 24

When we change the order of multiplication, the answer is (usually) different.

Example:

See how changing the order affects this multiplication:

Motrix 1 1 3 X 2 1 2

2
4
0
2
1×0+2×2
3×0+4×2
4
8

0
2
2
4

Motrix 1 1 3 X 2 3

2×2+0×4
1×2+2×4
4
10

The answers are different!

X 1 3 Times X 1 3

It can have the same result (such as when one matrix is the Identity Matrix) but not usually.