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Inverse Of Matrix

What Is The Need For Inverse?

Inverse means opposite of some operation performed and the result obtained is identity of that operation.

For example,

Additive identity
------------------
If a + 0 = a then 0 is additive identity.

3 + 0 = 3

but 

3 - 3 = 0 (additive identity) 

Therefore, subtraction is inverse operation of addition.

Multiplicative identity
-----------------------

If a \times 1 = a then 1 is multiplicative identity because it gives a as result.

4 \times 1 = 4 

but 

4 \times \frac{1}{4} = 1 ( multiplicative identity)

Therefore, multiplying with reciprocal or division is inverse operation of multiplication.

The same idea can be extended to Matrix since we are unable to divide two matrices directly. If A is a square matrix and invertible , then find an inverse matrix A^{-1} such that multiplying it with A will give an identity matrix I of same order.

A.A^{-1} = A^{-1}.A = I

Why Square Matrix ?

The inverse deals with negative power such as A^{-1}, a non-square matrix is cannot be used because it is undefined( cannot multiply).

The second reason for using square matrix is the identity matrix. An identity matrix is a square matrix only. A product of non-square matrix with its inverse will not result in an identity matrix.

If a square matrix A has inverse matrix B such that

AB = BA = I

Then the matrix A is called invertible matrix and matrix B is its inverse. If there is no B for matrix A, then it is called Singular matrix.

A matrix is singular and has no inverse if its determinant is 0. You will learn about determinants in future lessons.

Suppose A is a singular matrix of order 2 x 2.

A = \begin{bmatrix}a & b\\c & d\end{bmatrix}ad - bc = 0 

In the same manner, determinants of higher order matrices is found.

Therefore, only square matrix is used to find inverse which is also a square matrix of size n \times n.

Uniqueness Of Inverse Matrix

If a square matrix A is invertible, then it has exactly one inverse.

Proof :

Suppose that there are two inverse B and C for matrix A. We get

AB = BA = I  - (1)

AC = CA = I  - (2)

We know that any matrix multiplied by Identity matrix will result itself. Therefore, the following is true.

B = B.IB = B(AC) \\by (2)

B = (BA)C  \\ by associativity property

B = I.C  \\ by (1)

B = C

Therefore, inverse is unique.
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Inverse Of Matrix

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