Question

How to find the inverse of any matrix?​

Answer 1

Answer:

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Step-by-step explanation:

#The inverse of A is A-1 only when A × A-1 = A-1 × A = 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 inverse at all.

Answer 2

Step-by-step explanation:

This is the reciprocal of a number:

Reciprocal of 8 is 1/8 and back again

Reciprocal of a Number

The Inverse of a Matrix is the same idea but we write it A-1

Reciprocal of A is A-inverse and back again

Why not 1/A ? Because we don't divide by a matrix! And anyway 1/8 can also be written 8-1

And there are other similarities:

When we multiply a number by its reciprocal we get 1

8 × (1/8) = 1

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

A × A-1 = I

Same thing when the inverse comes first:

(1/8) × 8 = 1

A-1 × A = I

Identity Matrix

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

Identity Matrix

A 3x3 Identity Matrix

It is "square" (has same number of rows as columns),

It has 1s on the diagonal and 0s everywhere else.

Its symbol is the capital letter I.

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

Definition

Here is the definition:

The inverse of A is A-1 only when:

A × A-1 = A-1 × A = I

Sometimes there is no inverse at all.

2x2 Matrix

OK, how do we calculate the inverse?

Well, for a 2x2 matrix the inverse is:

matrix inverse 2x2 determinant

In other words: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc).

Let us try an example:

matrix inverse 2x2 ex1

How do we know this is the right answer?

Remember it must be true that: A × A-1 = I

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

matrix inverse 2x2 ex2

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

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

Why don't you have a go at multiplying these? See if you also get the Identity Matrix:

matrix inverse 2x2 ex3

Why Do 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 do I share 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.

The same thing can be done with matrices:

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

XA = B

It would be nice to 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)

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