We write A-1 instead of 1A because we don"t divide by a matrix!

And there are other similarities:


When we multiply a matrix by its inverse we get the Identity Matrix (which is lượt thích "1" for matrices):


Identity Matrix

We just mentioned the "Identity Matrix". It is the matrix equivalent of the number "1":


It is "square" (has same number of rows as columns),It has 1s on the diagonal & 0s everywhere else.Its symbol is the capital letter I.

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The Identity Matrix can be 2×2 in size, or 3×3, 4×4, etc ...

Definition

Here is the definition:


(Note: writing AA-1 means A times A-1)

2x2 Matrix

OK, how bởi vì we calculate the inverse?

Well, for a 2x2 matrix the inverse is:


In other words: swap the positions of a và d, put negatives in front of b and c, & divide everything by ad−bc .

Note: ad−bc is called the determinant.

Let us try an example:


How vì chưng we know this is the right answer?

Remember it must be true that: AA-1 = I

So, let us kiểm tra to see what happens when we multiply the matrix by its inverse:


And, hey!, we over up with the Identity Matrix! So it must be right.

It should also be true that: A-1A = I

Why don"t you have a go at multiplying these? See if you also get the Identity Matrix:


Why bởi We Need an Inverse?

Because with matrices we don"t divide! Seriously, there is no concept of dividing by a matrix.

But we can multiply by an inverse, which achieves the same thing.


Imagine we can"t divide by numbers ...

... And someone asks "How bởi vì I chia sẻ 10 apples with 2 people?"

But we can take the reciprocal of 2 (which is 0.5), so we answer:

10 × 0.5 = 5

They get 5 apples each.


Say we want lớn find matrix X, & we know matrix A và B:

XA = B

It would be nice lớn divide both sides by A (to get X=B/A), but remember we can"t divide.

But what if we multiply both sides by A-1 ?

XAA-1 = BA-1

And we know that AA-1 = I, so:

XI = BA-1

We can remove I (for the same reason we can remove "1" from 1x = ab for numbers):

X = BA-1

And we have our answer (assuming we can calculate A-1)


In that example we were very careful to get the multiplications correct, because with matrices the order of multiplication matters. AB is almost never equal khổng lồ BA.

A Real Life Example: Bus & Train

*

A group took a trip on a bus, at $3 per child và $3.20 per adult for a total of $118.40.

They took the train back at $3.50 per child và $3.60 per adult for a total of $135.20.

How many children, & how many adults?


First, let us set up the matrices (be careful to lớn get the rows and columns correct!):

*

This is just lượt thích the example above:

XA = B

So khổng lồ solve it we need the inverse of "A":


There were 16 children và 22 adults!

The answer almost appears lượt thích magic. But it is based on good mathematics.


Calculations lượt thích that (but using much larger matrices) help Engineers design buildings, are used in video games và computer animations to lớn make things look 3-dimensional, và many other places.

It is also a way to lớn solve Systems of Linear Equations.

The calculations are done by computer, but the people must understand the formulas.


Say that we are trying lớn find "X" in this case:

AX = B

This is different to lớn the example above! X is now after A.

With matrices the order of multiplication usually changes the answer. Vì not assume that AB = BA, it is almost never true.

So how vì chưng we solve this one? Using the same method, but put A-1 in front:

A-1AX = A-1B

And we know that A-1A= I, so:

IX = A-1B

We can remove I:

X = A-1B

And we have our answer (assuming we can calculate A-1)


Why don"t we try our bus and train example, but with the data phối up that way around.

It can be done that way, but we must be careful how we mix it up.

This is what it looks like as AX = B:


It looks so neat! I think I prefer it lượt thích this.

Also cảnh báo how the rows và columns are swapped over("Transposed") compared to lớn the previous example.

To solve it we need the inverse of "A":


It is lượt thích the inverse we got before, butTransposed (rows & columns swapped over).

Now we can solve using:

X = A-1B


Same answer: 16 children và 22 adults.

So matrices are powerful things, but they vị need khổng lồ be phối up correctly!

The Inverse May Not Exist

First of all, lớn have an inverse the matrix must be "square" (same number of rows và columns).

But also the determinant cannot be zero (or we over up dividing by zero). How about this:


24−24? That equals 0, and 1/0 is undefined.We cannot go any further! This matrix has no Inverse.

Such a matrix is called "Singular",which only happens when the determinant is zero.

And it makes sense ... Look at the numbers: the second row is just double the first row, & does not địa chỉ any new information.

And the determinant 24−24 lets us know this fact.

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(Imagine in our bus & train example that the prices on the train were all exactly 50% higher than the bus: so now we can"t figure out any differences between adults và children. There needs khổng lồ be something to lớn set them apart.)

Bigger Matrices

The inverse of a 2x2 is easy ... Compared to larger matrices (such as a 3x3, 4x4, etc).

For those larger matrices there are three main methods to lớn work out the inverse:

Conclusion


The inverse of A is A-1 only when AA-1 = A-1A = ITo find the inverse of a 2x2 matrix: swap the positions of a và d, put negatives in front of b & c, & divide everything by the determinant (ad-bc).Sometimes there is no inverse at all
Multiplying Matrices Determinant of a Matrix Matrix Calculator Algebra Index