Changed of a Matrix

Please read our Introduction to Matrices kickoff.

What is the Inverse of a Matrix?

Just like a number has a reciprocal ...

Reciprocal of 8 is 1/8 and back again
Reciprocal of a Number (note: ane 8 tin likewise be written 8-1 )

... a matrix has an inverse :

Reciprocal of A is A-inverse and back again
Changed of a Matrix

We write A-1 instead of i A   because we don't divide by a matrix!

And in that location are other similarities:

When we multiply a number by its reciprocal we go 1:

viii × one 8 = one

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

A × A-ane = I

Same matter when the inverse comes first:

one 8 × eight = 1

A-1 × A = I

Identity Matrix

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

I =

A 3x3 Identity Matrix

  • Information technology is "square" (has same number of rows as columns),
  • It has onesouth on the diagonal and 0south everywhere else.
  • Its symbol is the capital letter letter I.

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

Definition

Hither is the definition:

The inverse of A is A-1 merely when:

AA-1 = A-1A = I

Sometimes there is no inverse at all.

(Note: writing AA-ane ways A times A-1)

2x2 Matrix

OK, how practice nosotros calculate the inverse?

Well, for a 2x2 matrix the inverse is:

In other words: bandy the positions of a and d, put negatives in forepart of b and c, and carve up everything by ad−bc .

Note: advertisement−bc is chosen the determinant.

Permit us try an example:



How do we know this is the right respond?

Remember information technology must be true that: AA-ane = I

Then, let united states check to see what happens when nosotros multiply the matrix by its inverse:

=

4×0.6+vii×−0.2 iv×−0.7+7×0.4 2×0.6+half dozen×−0.2 ii×−0.7+6×0.iv


=

2.4−one.4 −2.eight+2.8 i.ii−1.ii −i.4+2.iv


And, hey!, we end up with the Identity Matrix!
So it must be correct.

It should also exist true that: A-iA = I

Why don't y'all have a go at multiplying these? Come across if you also get the Identity Matrix:

Why Do Nosotros Demand an Changed?

Because with matrices we don't carve up! Seriously, there is no concept of dividing by a matrix.

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

Imagine we tin't carve up by numbers ...

... and someone asks "How do I share x apples with 2 people?"

Just we can take the reciprocal of two (which is 0.5), so we respond:

10 × 0.5 = 5

They become 5 apples each.

The same thing tin be done with matrices:

Say nosotros want to find matrix X, and we know matrix A and B:

XA = B

Information technology would be overnice to carve up both sides by A (to become Ten=B/A), just think we can't divide.

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

XAA-ane = BA-1

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

XI = BA-1

Nosotros tin can remove I (for the same reason nosotros can remove "1" from 1x = ab for numbers):

X = BA-i

And we have our reply (bold nosotros tin calculate A-i)

In that example we were very conscientious to go the multiplications right, because with matrices the order of multiplication matters. AB is almost never equal to BA.

A Real Life Instance: Bus and Railroad train

A group took a trip on a autobus, at $3 per kid and $3.xx per developed for a total of $118.forty.

They took the train back at $3.50 per kid and $three.lx per developed for a total of $135.twenty.

How many children, and how many adults?

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

matrix inverse 2x2 bus

This is but similar the case to a higher place:

XA = B

So to solve it we need the inverse of "A":


At present we accept the inverse nosotros can solve using:

10 = BA-one


=

118.iv×−ix + 135.2×eight 118.4×viii.75 + 135.2×−seven.5


There were sixteen children and 22 adults!

The answer nearly appears like magic. But information technology is based on expert mathematics.

Calculations like that (simply using much larger matrices) assist Engineers design buildings, are used in video games and figurer animations to make things wait iii-dimensional, and many other places.

Information technology is besides a way to solve Systems of Linear Equations.

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

Order is Important

Say that nosotros are trying to find "10" in this case:

AX = B

This is unlike to the example to a higher place! X is now later A.

With matrices the club of multiplication unremarkably changes the answer. Do not assume that AB = BA, it is almost never true.

And so how do we solve this one? Using the same method, but put A-one in front end:

A-1AX = A-oneB

And we know that A-aneA= I, then:

IX = A-1B

We tin can remove I:

X = A-1B

And nosotros take our respond (assuming we can calculate A-1)

Why don't we try our bus and train example, merely with the data fix up that style around.

Information technology can exist done that style, just nosotros must exist careful how nosotros set it upward.

This is what it looks like as AX = B:

It looks so neat! I think I prefer it similar this.

Besides note how the rows and columns are swapped over
("Transposed") compared to the previous example.

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


It is like the inverse nosotros got before, but
Transposed (rows and columns swapped over).

Now we tin solve using:

X = A-1B


=

−9×118.iv + viii×135.2 8.75×118.4 − 7.five×135.ii


Aforementioned answer: sixteen children and 22 adults.

So matrices are powerful things, but they do need to be ready correctly!

The Changed May Not Exist

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

But too the determinant cannot be zero (or we stop upwardly dividing by zero). How most this:


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

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

And it makes sense ... expect at the numbers: the 2nd row is just double the first row, and does not add any new information.

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

(Imagine in our jitney and railroad train example that the prices on the train were all exactly 50% higher than the bus: so now we tin't figure out any differences between adults and children. In that location needs to be something to 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 work out the inverse:

  • Inverse of a Matrix using Elementary Row Operations (Gauss-Jordan)
  • Inverse of a Matrix using Minors, Cofactors and Adjugate
  • Utilise a reckoner (such as the Matrix Computer)

Determination

  • The inverse of A is A-one but when AA-1 = A-1A = I
  • To find the inverse of a 2x2 matrix: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc).
  • Sometimes there is no changed at all