Question

. [Stacked matrix] (/10)Let A be anm×nmatrix, and consider the stacked matrix S definedby S =(AI)When does S have linearly independent columns? When does S havelinearly independent rows? Your answer can depend on m, n, orwhether or not A has linearly independent columns or rows.

Answer 1

S has n linearly independent rows, but not all the rows are independent.

Explanation:

From the information provided:

Let A be an ( n × m ) matrix, and the stacked matrix is defined by:

`S = \left[\begin{array}{c}A&\\I\end{array}\right]`

Then I will be ( n × n) matrix.

So, S will have m + n integers in the row and n integers in the column.

Hence, S must be (m + n) n matrix

From the matrix; we can posit that S has n linearly independent column vectors and n linearly independent rows.

Similarly, 'I' have n linearly independent rows.

{ row vector of S} = { row vector of I} U { row vector of A}

Therefore, S has n linearly independent rows, but not all the rows are independent.