Question

Let A be a singular row reduced echelon square matrix of size n .Find a matr xA(x)=(a_(ij)(x))_(n times n) where a_(ij)(x)' s are polynomials in x such that A(0)= (a_(ij)(0))_(n times n)=A but A(delta)=(a_(ij)(delta))_(n times n) is nonsingular for all delta!=0.

Answer 1

If the reduced row echelon form of a square matrix is not the identity then it can never be Non - singular

We will consider a real example below

$A=\begin{bmatrix} 1 & 2 & -2 \\ 2 & -3 & 10\\ 2 & -3 & 10 \end{bmatrix}$ This matrix can be reduced in echelon square form.

$A=\begin{bmatrix} 1 & 2 & -2 \\ 0 & -7 & 6\\ 0 & -7 & 14 \end{bmatrix}$ $R_2 \to R_2-2R_1$ and $R_3 \to R_3-2R_1$

$A=\begin{bmatrix} 1 & 2 & -2 \\ 0 & -7 & 6\\ 0 & 0 & 8 \end{bmatrix}$ $R_3 \to R_3-R_2$

So here A has three pivot columns 1,-7,8. All the columns are linearly independent

$Pan \space k (A)=3 \implies |A|\not =0$

Therefore, A is non-singular.