## Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)

## Problem 136

Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.

Add to solve laterLet $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.

Add to solve laterLet $A$ 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 $\rk(AB)=\rk(A)$.

Find **a square root** of the matrix

\[A=\begin{bmatrix}

1 & 3 & -3 \\

0 &4 &5 \\

0 & 0 & 9

\end{bmatrix}.\]

How many square roots does this matrix have?

(*University of California, Berkeley Qualifying Exam*)

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Let

\[A=\begin{bmatrix}

1 & 1 & 0 \\

1 &1 &0

\end{bmatrix}\]
be a matrix.

Find a basis of the null space of the matrix $A$.

(Remark: a null space is also called a kernel.)

Add to solve laterLet $A$ be the following $3 \times 3$ matrix.

\[A=\begin{bmatrix}

1 & 1 & -1 \\

0 &1 &2 \\

1 & 1 & a

\end{bmatrix}.\]
Determine the values of $a$ so that the matrix $A$ is nonsingular.

Let $A$ be an $m \times n$ real matrix. Then the **null space** $\calN(A)$ of $A$ is defined by

\[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\]
That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$.

Prove that the null space $\calN(A)$ is a subspace of the vector space $\R^n$.

(Note that the null space is also called the **kernel** of $A$.)

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Express the vector $\mathbf{b}=\begin{bmatrix}

2 \\

13 \\

6

\end{bmatrix}$ as a linear combination of the vectors

\[\mathbf{v}_1=\begin{bmatrix}

1 \\

5 \\

-1

\end{bmatrix},

\mathbf{v}_2=

\begin{bmatrix}

1 \\

2 \\

1

\end{bmatrix},

\mathbf{v}_3=

\begin{bmatrix}

1 \\

4 \\

3

\end{bmatrix}.\]

(*The Ohio State University, Linear Algebra Exam*)

Let

\[A=\begin{bmatrix}

-1 & 2 \\

0 & -1

\end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix}

1\\

0

\end{bmatrix}.\]
Compute $A^{2017}\mathbf{u}$.

(*The Ohio State University, Linear Algebra Exam*)

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Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.

**(a)** The product $AB$ is symmetric if and only if $AB=BA$.

**(b)** If the product $AB$ is a diagonal matrix, then $AB=BA$.

Test your understanding of basic properties of matrix operations.

There are **10 True or False Quiz Problems**.

These 10 problems are very common and essential.

So make sure to understand these and don’t lose a point if any of these is your exam problems.

(These are actual exam problems at the Ohio State University.)

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.

Click the **View question** button to see the solutions.

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*)

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Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.

**(a)** \[\left\{

\begin{array}{c}

ax+by=c \\

dx+ey=f,

\end{array}

\right.

\]
where $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.

**(b)** $A \mathbf{x}=\mathbf{0}$, where $A$ is a singular matrix.

**(c)** A homogeneous system of $3$ equations in $4$ unknowns.

**(d) **$A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:

\[\begin{bmatrix}

1 & 0 & -1 & 0 \\

0 &1 & 2 & 0 \\

0 & 0 & 0 & 1

\end{bmatrix}.\]
(*The Ohio State University, Linear Algebra Exam*)

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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}.\]

(*Johns Hopkins University, Linear Algebra Exam*)

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Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix.

Is it true that the matrix product with opposite order $BA$ is also the zero matrix?

If so, give a proof. If not, give a counterexample.

Let $A$ and $B$ be $2\times 2$ matrices.

Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2-B^2$.

Add to solve laterConsider a polynomial

\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.

Define the matrix

\[A=\begin{bmatrix}

0 & 0 & \dots & 0 &-a_0 \\

1 & 0 & \dots & 0 & -a_1 \\

0 & 1 & \dots & 0 & -a_2 \\

\vdots & & \ddots & & \vdots \\

0 & 0 & \dots & 1 & -a_{n-1}

\end{bmatrix}.\]

Then prove that the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$.

The matrix is called the ** companion matrix** of the polynomial $p(x)$.

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Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix.

Show that the matrix $A$ is diagonalizable.

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Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.

That is,

\begin{equation*}

V:=\left\{ A=\begin{bmatrix}

a_{11} & 0 & \dots & 0 \\

0 &a_{22} & \dots & 0 \\

0 & 0 & \ddots & \vdots \\

0 & 0 & \dots & a_{nn}

\end{bmatrix} \quad \middle| \quad

\begin{array}{l}

a_{11}, \dots, a_{nn} \in \C,\\

\tr(A)=0 \\

\end{array}

\right\}

\end{equation*}

Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.

**(a)** Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)

**(b)** Show that matrices

\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.

**(c)** Find the dimension of $V$.

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Let A be the matrix

\[\begin{bmatrix}

1 & -1 & 0 \\

0 &1 &-1 \\

0 & 0 & 1

\end{bmatrix}.\]
Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.

(*The Ohio State University Linear Algebra Exam*)