## Question

Let be a non-singular matrix. If column is replaced by a and that the resulting matrix is called along with , then state the necessary and sufficient condition for to be non-singular.

## Properties for the Solution

Since B is a non-singular matrix composed of column vectors, it satisfies the following properties:

1. B is a square matrix with size .

2. Transpose of B will not change its non-singularity nature or rank.

3. B has full rank ⟹ rank(B)=rank()=r=n

4. is a linearly independent system of rank n.

## Solution

Now, we can say that BT is a matrix with row vectors and say, column vectors .

Row vectors can be written as:

As per question, . This means that,

Assume that is a scalar quantity. Our aim is to replace with a. Let’s say we did that.

Using elementary row operations, if we subtract a with etc., a will become a zero vector. If that happens, the rank of will become . This will cause the system to be linearly dependent & lose its non-singular nature.

This can be avoided if is never equal to zero. In that case, the term will remain non-zero within the system after elementary row operations.

**Thus, the necessary & sufficient condition for to be non-singular is .**