What Is The Need For Inverse?
Inverse means opposite of some operation performed and the result obtained is identity of that operation.
Additive identity ------------------ If then is additive identity. but (additive identity) Therefore, subtraction is inverse operation of addition. Multiplicative identity ----------------------- If then is multiplicative identity because it gives as result. but ( 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 is a square matrix and invertible , then find an inverse matrix such that multiplying it with will give an identity matrix of same order.
Why Square Matrix ?
The inverse deals with negative power such as , 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 has inverse matrix such that
Then the matrix is called invertible matrix and matrix is its inverse. If there is no for matrix , 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 is a singular matrix of order 2 x 2. 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 .
Uniqueness Of Inverse Matrix
If a square matrix is invertible, then it has exactly one inverse.
Suppose that there are two inverse and for matrix . We get
- (1) - (2)
We know that any matrix multiplied by Identity matrix will result itself. Therefore, the following is true.
\\by (2) \\ by associativity property \\ by (1) Therefore, inverse is unique.
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