Properties of Inverse Matrix Proof

What is Inverse Matrix?

Let A be a square matrix of order n. A matrix B of order n is said to be the inverse of A if AB=BA=I_n

Note:

The inverse of an arbitrary square matrix may not exist.

Example of Inverse Matrix ?

The Matrix A=\;\begin{bmatrix}1&-1\\-1&1\end{bmatrix} has no inverse.

An inverse of a matrix, if it exists, is unique. For suppose that A has two inverses, say B and C. Then

AB = BA=I and AC = CA= I

Therefore, by the associative law for multiplication, we have

B(AC) = BI = B

and (BA)C = IC = C

and so B = C

The unique inverse of a matrix A, if it exists, is denoted by A^{-1}

A square matrix A, whose inverse exists, is called a nonsingular or invertible matrix. Square matrices which do not have inverse are called singular matrices.

Properties of Inverse Matrix Proof

It is easy to verify the following properties of inverses of matrices:

  • For any invertible matrix A,(A^{-1})^{-1} = A
  • For nonsingular matrices A and B, (AB)^{-1} = B^{-1} A^{-1}
  • For any invertible matrix A, (A^t)^{-1} = (A^{-1})^t

Inverse Matrix Proof

For nonsingular matrices A and B, (AB)^{-1} = B^{-1} A^{-1}.

Since A and B are nonsingular, A^{-1} and B^{-1} exist. Also, Since A and B square matrices AB are defined.

To prove that AB is nonsingular and B^{-1} A^{-1} is the inverse of AB, We show that

(AB)

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