91影视

Let 鈥楢鈥� be an m x n matrix from$R^n鈫�R^m$then the image of the matrix A, denoted by Im(A), is defined as

Im (A) = span of the columns of A

$鈬�Im\left(A\right)=A\left(R^n\right)鈯�R^m$

Let us suppose that A is a matrix of order n x n from$R^n鈫�R^n$which is not invertible. Then rank of the matrix A is less than 鈥榥鈥�.

Thus, the solution of the system of the equation AX = B will be inconsistent and the system of the equation AX = B has no solution.

It means that 鈥楤鈥� doesn鈥檛 lie in the range $R^n$of A.

Consequently, Im(A) cannot be equal to $R^n$as it doesn鈥檛 contain 鈥楤鈥�, which contradicts the fact that the image of an n x n matrix A is equal to $R^n$.

Thus, our supposition is wrong.

Hence, matrix A of order n x n must be invertible.

If the image of an n x n matrix A is all of$R^n$ , then A must be invertible.