# Tagged: elementary row operations

## Problem 569

For an $m\times n$ matrix $A$, we denote by $\mathrm{rref}(A)$ the matrix in reduced row echelon form that is row equivalent to $A$.
For example, consider the matrix $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &2 &2 \end{bmatrix}$
Then we have
$A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &2 &2 \end{bmatrix} \xrightarrow{\frac{1}{2}R_2} \begin{bmatrix} 1 & 1 & 1 \\ 0 &1 & 1 \end{bmatrix} \xrightarrow{R_1-R_2} \begin{bmatrix} 1 & 0 & 0 \\ 0 &1 &1 \end{bmatrix}$ and the last matrix is in reduced row echelon form.
Hence $\mathrm{rref}(A)=\begin{bmatrix} 1 & 0 & 0 \\ 0 &1 &1 \end{bmatrix}$.

Find an example of matrices $A$ and $B$ such that
$\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B).$

## Problem 552

For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.

(a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ 0 & 1 & 1 \end{bmatrix}$

(b) $A=\begin{bmatrix} 1 & 0 & 2 \\ -1 &-3 &2 \\ 3 & 6 & -2 \end{bmatrix}$.

## Problem 442

Consider the following system of linear equations
\begin{align*}
2x+3y+z&=-1\\
3x+3y+z&=1\\
2x+4y+z&=-2.
\end{align*}

(a) Find the coefficient matrix $A$ for this system.

(b) Find the inverse matrix of the coefficient matrix found in (a)

(c) Solve the system using the inverse matrix $A^{-1}$.

## Problem 400

Find all the eigenvalues and eigenvectors of the matrix
$A=\begin{bmatrix} 10001 & 3 & 5 & 7 &9 & 11 \\ 1 & 10003 & 5 & 7 & 9 & 11 \\ 1 & 3 & 10005 & 7 & 9 & 11 \\ 1 & 3 & 5 & 10007 & 9 & 11 \\ 1 &3 & 5 & 7 & 10009 & 11 \\ 1 &3 & 5 & 7 & 9 & 10011 \end{bmatrix}.$

(MIT, Linear Algebra Homework Problem)

## Problem 324

Let $T$ be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations.
\begin{align*}
T\left(\, \begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix} \,\right)
=\begin{bmatrix}
1 \\
0 \\
1
2 \\
3 \\
5
\end{bmatrix} \, \right) =
\begin{bmatrix}
0 \\
2 \\
-1
T \left( \, \begin{bmatrix}
0 \\
1 \\
2
\end{bmatrix} \, \right)=
\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}.
\end{align*}
Then for any vector
$\mathbf{x}=\begin{bmatrix} x \\ y \\ z \end{bmatrix}\in \R^3,$ find the formula for $T(\mathbf{x})$.

## Problem 320

(a) Let $A=\begin{bmatrix} 1 & 3 & 0 & 0 \\ 1 &3 & 1 & 2 \\ 1 & 3 & 1 & 2 \end{bmatrix}$.
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.

(b) Find the rank and nullity of the matrix $A$ in part (a).

## Problem 313

(a) Let $A=\begin{bmatrix} 1 & 2 & 1 \\ 3 &6 &4 \end{bmatrix}$ and let
$\mathbf{a}=\begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix} 1 \\ 1 \end{bmatrix}.$ For each of the vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, determine whether the vector is in the null space $\calN(A)$. Do the same for the range $\calR(A)$.

(b) Find a basis of the null space of the matrix $B=\begin{bmatrix} 1 & 1 & 2 \\ -2 &-2 &-4 \end{bmatrix}$.

## Problem 303

Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors
$\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}, \text{ and } \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}.$ Then find the rank of the matrix $A$.

(Purdue University, Linear Algebra Final Exam Problem)

## Problem 299

Find the inverse matrix of
$A=\begin{bmatrix} 1 & 1 & 2 \\ 0 &0 &1 \\ 1 & 0 & 1 \end{bmatrix}$ if it exists. If you think there is no inverse matrix of $A$, then give a reason.

(The Ohio State University, Linear Algebra Midterm Exam Problem)

## Problem 289

(a) Find the inverse matrix of
$A=\begin{bmatrix} 1 & 0 & 1 \\ 1 &0 &0 \\ 2 & 1 & 1 \end{bmatrix}$ if it exists. If you think there is no inverse matrix of $A$, then give a reason.

(b) Find a nonsingular $2\times 2$ matrix $A$ such that
$A^3=A^2B-3A^2,$ where
$B=\begin{bmatrix} 4 & 1\\ 2& 6 \end{bmatrix}.$ Verify that the matrix $A$ you obtained is actually a nonsingular matrix.

(The Ohio State University, Linear Algebra Midterm Exam Problem)

## Problem 280

Determine whether there exists a nonsingular matrix $A$ if
$A^2=AB+2A,$ where $B$ is the following matrix.
If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

(a) $B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 1 & 2 & -2 \end{bmatrix}$

(b) $B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.$

## Problem 277

Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent, express one vector in the set as a linear combination of the others.
$\left\{\, \begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}, \begin{bmatrix} -1 \\ -2 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} -2 \\ -2 \\ 7 \\ 11 \end{bmatrix}\, \right\}.$

## Problem 276

Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients.
Let $T: P_3 \to V$ be the linear transformation defined by
$T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix} a_0+a_2 & -a_0+a_3\\ a_1-a_2 & -a_1-a_3 \end{bmatrix}$ for any polynomial $a_0+a_1x+a_2x^2+a_3 \in P_3$.
Find a basis for the range of $T$, $\calR(T)$, and determine the rank of $T$, $\rk(T)$, and the nullity of $T$, $\nullity(T)$.
Also, prove that $T$ is not injective.

## Problem 275

Let $A$ be the following $3\times 3$ upper triangular matrix.
$A=\begin{bmatrix} 1 & x & y \\ 0 &1 &z \\ 0 & 0 & 1 \end{bmatrix},$ where $x, y, z$ are some real numbers.

Determine whether the matrix $A$ is invertible or not. If it is invertible, then find the inverse matrix $A^{-1}$.

## Problem 272

Let
$A=\begin{bmatrix} 1 & 3\\ 2& 4 \end{bmatrix}.$ Then

(a) Find all matrices
$B=\begin{bmatrix} x & y\\ z& w \end{bmatrix}$ such that $AB=BA$.

(b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that
$AB=BA \text{ and } AC \neq CA.$

## Problem 267

Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination).

Find the vector form for the general solution.
\begin{align*}
x_1-x_3-3x_5&=1\\
3x_1+x_2-x_3+x_4-9x_5&=3\\
x_1-x_3+x_4-2x_5&=1.
\end{align*}

## Problem 262

(a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed.
\begin{align*}
x_1+x_2-x_5&=1\\
x_2+2x_3+x_4+3x_5&=1\\
x_1-x_3+x_4+x_5&=0
\end{align*}

(b) Determine all possibilities for the solution set of a homogeneous system of $2$ equations in $2$ unknowns that has a solution $x_1=1, x_2=5$.

## Problem 260

Let $A=\begin{bmatrix} 1 & 1 & 2 \\ 2 &2 &4 \\ 2 & 3 & 5 \end{bmatrix}.$

(a) Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$.

(b) Find a basis for the null space of $A$.

(c) Find a basis for the range of $A$ that consists of columns of $A$. For each columns, $A_j$ of $A$ that does not appear in the basis, express $A_j$ as a linear combination of the basis vectors.

(d) Exhibit a basis for the row space of $A$.

## Problem 248

We say that two $m\times n$ matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations.

Let $A$ and $I$ be $2\times 2$ matrices defined as follows.
$A=\begin{bmatrix} 1 & b\\ c& d \end{bmatrix}, \qquad I=\begin{bmatrix} 1 & 0\\ 0& 1 \end{bmatrix}.$ Prove that the matrix $A$ is row equivalent to the matrix $I$ if $d-cb \neq 0$.

$A=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ -1 & -3 & -4 \end{bmatrix} \text{ and } B=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ 5 & 3 & 3 \end{bmatrix}.$
Determine the null spaces of matrices $A$ and $B$.