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Questions for Math-1: A course for Biology and Applied Geology students

Abhay Soman

Exercises 1.4 Exercises

A few exercises related to the row reduced echelon form of a matrix are listed below.

1.

Find the row reduced echelon form of the following matrices.
\begin{equation*} \begin{pmatrix}1\amp 2\amp -1\amp 2\\0\amp 0\amp 0\amp 0\\3\amp 1\amp -2\amp 0\end{pmatrix};\quad\begin{pmatrix}0\amp 1\amp -1\\0\amp -4\amp -1\\3\amp 3\amp 3\\-1\amp -1\amp -1\\2\amp 2\amp 2\end{pmatrix};\quad\begin{pmatrix}0\amp 1\\0\amp 0\end{pmatrix}\text{.} \end{equation*}

2.

How many pivots are there in one row? How many pivots are there in one column?

3.

In a \(3\times 10\) matrix what is the largest possible number of pivots?

4.

Write the row reduced echelon matrix whose every row has a pivot.

5.

Consider the following augmented matrices. In each case consider the corresponding system of linear equations and determine whether the system has no solution, a unique solution, or infinitely many solutions.
  1. \begin{equation*} \left( \begin{matrix}1\amp 0\amp 0\\0\amp 1\amp 0\\0\amp 0\amp 1\end{matrix}\left|\,\begin{matrix}4\\-1\\2\end{matrix}\right. \right) \end{equation*}
  2. \begin{equation*} \left( \begin{matrix}1\amp 0\amp 1\\0\amp 1\amp 1\\0\amp 0\amp 0\end{matrix}\left|\,\begin{matrix}-2\\4\\0\end{matrix}\right. \right) \end{equation*}
  3. \begin{equation*} \left( \begin{matrix}1\amp 8\amp 1\\0\amp 1\amp 1\\0\amp 0\amp 0\end{matrix}\left|\,\begin{matrix}-2\\4\\1\end{matrix}\right. \right) \end{equation*}
  4. \begin{equation*} \left(\begin{matrix}i\amp 0\amp 0\amp 0\\1\amp -i\amp 0\amp 0\\1\amp 2\amp i\amp 0\\1\amp 3\amp 3\amp -i\end{matrix}\left|\,\begin{matrix}1\\2\\3\\3\end{matrix}\right.\right), \end{equation*}
    where \(i^2=-1\text{.}\)

6.

Consider any square matrix of size \(n\) of your choice. Using the row reduced echelon form determine whether your matrix is invertible or not (see Definition 1.1.12).

7.

List all possible types of elementary matrices of size \(2\) and \(3\text{.}\) Further show that these matrices are invertible.

8.

Let \(A\in M_3(\K)\text{.}\) Perform the row operation described in Item 3, and denote the resulting matrix by \(A^\prime\text{.}\) Now perform the exact same operation on the identity matrix \(I_3\in M_3(\K)\text{,}\) and denote the resulting matrix by \(E\) (note that \(E\) is an elemetary matrix). Show that \(EA=A^\prime\text{.}\) What happens if we consider \(AE\) in terms of columns of \(A\text{?}\)
Do the same exercise for other two row operations mentioned in Definition 1.3.1.

9.

Consider the system of equations \(AX=B\) with \(A\in M_{m\times n}(\R)\) and \(B\in M_{m\times 1}(\R)\text{.}\)
  1. Show that if the above system has two distinct solutions then it has infinitely many solutions.
  2. Show that the above system has a solution in the complex numbers if and only if it has a solution in the real numbers.

10.

Let \(A\in M_n(\K)\text{.}\) Show that if the system \(AX=B\) has a unique solution for some particular \(B\in M_{n\times 1}(\K)\) then the system \(AX=C\) has a unique solution for any \(C\in M_{n\times 1}(\K)\text{.}\)
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