# Tagged: rank of a matrix

## Problem 671

Determine whether the following matrices are nonsingular or not.

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

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

## Problem 670

Determine the values of a real number $a$ such that the matrix
$A=\begin{bmatrix} 3 & 0 & a \\ 2 &3 &0 \\ 0 & 18a & a+1 \end{bmatrix}$ is nonsingular.

## Problem 646

(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.

(b) Find all such matrices with rank 2.

## Problem 645

Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix.

Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.

## Problem 644

If $A, B$ have the same rank, can we conclude that they are row-equivalent?

If so, then prove it. If not, then provide a counterexample.

## Problem 643

For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.

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

(b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$.

(c) $C = \begin{bmatrix} 2 & -2 & 4 \\ 4 & 1 & -2 \\ 6 & -1 & 2 \end{bmatrix}$.

(d) $D = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}$.

(e) $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$.

## Problem 484

Let $A$ be a square matrix and its characteristic polynomial is given by
$p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).$ Find the rank of $A$.

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

## Problem 441

Let $A$ and $B$ be $m\times n$ matrices.
Prove that
$\rk(A+B) \leq \rk(A)+\rk(B).$

## Problem 352

A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors
$\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\in \R^n$ satisfying the linear equation of the form
$a_1x_1+a_2x_2+\cdots+a_nx_n=b,$ where $a_1, a_2, \dots, a_n$ (at least one of $a_1, a_2, \dots, a_n$ is nonzero) and $b$ are real numbers.
Here at least one of $a_1, a_2, \dots, a_n$ is nonzero.

Consider the hyperplane $P$ in $\R^n$ described by the linear equation
$a_1x_1+a_2x_2+\cdots+a_nx_n=0,$ where $a_1, a_2, \dots, a_n$ are some fixed real numbers and not all of these are zero.
(The constant term $b$ is zero.)

Then prove that the hyperplane $P$ is a subspace of $R^{n}$ of dimension $n-1$.

## Problem 329

Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.
Prove the followings.

(a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.

(b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the nullspace $\calN(T)$ of $T$.
Let $\mathbf{w}$ be the $n$-dimensional vector that is not in $\calN(T)$. Then
$B’=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}, \mathbf{w}\}$ is a basis of $\R^n$.

(c) Each vector $\mathbf{u}\in \R^n$ can be expressed as
$\mathbf{u}=\mathbf{v}+\frac{T(\mathbf{u})}{T(\mathbf{w})}\mathbf{w}$ for some vector $\mathbf{v}\in \calN(T)$.

## 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 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 295

Determine all possibilities for the number of solutions of each of the system of linear equations described below.

(a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution.

(b) A homogeneous system of $5$ equations in $4$ unknowns and the rank of the system is $4$.

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

## Problem 269

Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$.
Then prove the following statements.

(a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.

(b) The rank of $A$ is even.

## Problem 217

Let $A, B, C$ are $2\times 2$ diagonalizable matrices.

The graphs of characteristic polynomials of $A, B, C$ are shown below. The red graph is for $A$, the blue one for $B$, and the green one for $C$.

From this information, determine the rank of the matrices $A, B,$ and $C$.

Graphs of characteristic polynomials

## Problem 154

Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\right )=\begin{bmatrix} x_1-x_2 \\ x_1+x_2 \\ x_2 \end{bmatrix}$.

(a) Show that $T$ is a linear transformation.

(b) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.

(c) Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.

## Problem 140

Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$.
The dimension of the nullspace of $A$ is called the nullity of $A$.
Prove the followings.

(a) $\calN(A)=\calN(A^{\trans}A)$.

(b) $\rk(A)=\rk(A^{\trans}A)$.

## Problem 135

Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.

(a) $\rk(AB) \leq \rk(A)$.

(b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.

## Problem 103

Find the rank of the following real matrix.
$\begin{bmatrix} a & 1 & 2 \\ 1 &1 &1 \\ -1 & 1 & 1-a \end{bmatrix},$ where $a$ is a real number.

(Kyoto University, Linear Algebra Exam)
For which choice(s) of the constant $k$ is the following matrix invertible?
$A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &k \\ 1 & 4 & k^2 \end{bmatrix}.$