Question

let A be a square matrix of order 3 with eigen value2 , 2 and 3. A is diagonalizable then find rank of (A-2i)​

Answer 1

Let A be a square matrix of order 3 with eigen values 2, 2, 3. As A is diagonalizable then the rank of (A - 2i) is 2

1) Since A is 3 * 3 matrix, and is diagonalizable, the diagonal elements of the matrix are 2, 2, 3 (eigen values).

2) The matrix can be represented as [2   x   y]

                                                             [0   2   z]

                                                             [0   0   3] , where x, y, z are any random real numbers.

3) So, ( A - 2i ) will become [0   x   y]

                                             [0   0   z]

                                             [0   0   1]

4) As we can see here one of the column has become 0. So, the rank of the resulting matrix is 2.