Question

An elementary ntimesn row replacement matrix is the same as the ntimesn identity matrix with at least one of the 1's replaced with some number k. This means it is a triangular matrix, and so its determinant is the product of its diagonal entries.​ Thus, the determinant of an elementary row replacement matrix is __________

options are ;

1.exactly one, all , at least one

2. 1's or 0's

3.Identity matrix,invertible matrix , triangular matrix or a zero matrix

4. product or sum

5. a number

Answer 1

1's or 0's

Explanation:

Thinking process:

The matrix property: det (AB) = det (A) det (B)

Adding the multiple of one row to another will be equivalent to left multiplication by an elementary matrix.

For example, let E be some form of matrix such that:

n x n matrix, and so E is an n x n elementary matrix which acts as an operator which adds l copies to the i row and to row j.

Applying the same row operation to B results in the matrix AB.

This, the matrix, without the loss of generality becomes:

`\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]`

therefore, as seen from the matrix, the triangular matrix of the product is diagonal.

The matrix will have a diagonal so the determint of A, det A = 1

thus: det (AB) = det (A)det(B) = 1 det (B) = det (B)