## The Transpose of a Nonsingular Matrix is Nonsingular

## Problem 558

Let $A$ be an $n\times n$ nonsingular matrix.

Prove that the transpose matrix $A^{\trans}$ is also nonsingular.

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Let $A$ be an $n\times n$ nonsingular matrix.

Prove that the transpose matrix $A^{\trans}$ is also nonsingular.

Add to solve later Let $\mathbf{v}$ be a nonzero vector in $\R^n$.

Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$.

Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by

\[A=I-a\mathbf{v}\mathbf{v}^{\trans},\]
where $I$ is the $n\times n$ identity matrix.

Prove that $A$ is a symmetric matrix and $AA=I$.

Conclude that the inverse matrix is $A^{-1}=A$.

Let $A$ be a square matrix.

Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$.

Namely, show that

\[(A^{\trans})^{-1}=(A^{-1})^{\trans}.\]

Let $A$ be a square matrix such that

\[A^{\trans}A=A,\]
where $A^{\trans}$ is the transpose matrix of $A$.

Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.

Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that

\[\mathbf{y}A=\mathbf{y}.\]
(Here a row vector means a $1\times n$ matrix.)

Prove that there is a nonzero column vector $\mathbf{x}$ such that

\[A\mathbf{x}=\mathbf{x}.\]
(Here a column vector means an $n \times 1$ matrix.)

Suppose $A$ is a positive definite symmetric $n\times n$ matrix.

**(a)** Prove that $A$ is invertible.

**(b)** Prove that $A^{-1}$ is symmetric.

**(c)** Prove that $A^{-1}$ is positive-definite.

(*MIT, Linear Algebra Exam Problem*)

Read solution

Suppose that $A$ is a real $n\times n$ matrix.

**(a)** Is it true that $A$ must commute with its transpose?

**(b)** Suppose that the columns of $A$ (considered as vectors) form an orthonormal set.

Is it true that the rows of $A$ must also form an orthonormal set?

(*University of California, Berkeley, Linear Algebra Qualifying Exam*)

Let $A, B, C$ be the following $3\times 3$ matrices.

\[A=\begin{bmatrix}

1 & 2 & 3 \\

4 &5 &6 \\

7 & 8 & 9

\end{bmatrix}, B=\begin{bmatrix}

1 & 0 & 1 \\

0 &3 &0 \\

1 & 0 & 5

\end{bmatrix}, C=\begin{bmatrix}

-1 & 0\ & 1 \\

0 &5 &6 \\

3 & 0 & 1

\end{bmatrix}.\]
Then compute and simplify the following expression.

\[(A^{\trans}-B)^{\trans}+C(B^{-1}C)^{-1}.\]

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

Read solution

**(a)** The given matrix is the augmented matrix for a system of linear equations.

Give the vector form for the general solution.

\[ \left[\begin{array}{rrrrr|r}

1 & 0 & -1 & 0 &-2 & 0 \\

0 & 1 & 2 & 0 & -1 & 0 \\

0 & 0 & 0 & 1 & 1 & 0 \\

\end{array} \right].\]

**(b)** Let

\[A=\begin{bmatrix}

1 & 2 & 3 \\

4 &5 &6

\end{bmatrix}, B=\begin{bmatrix}

1 & 0 & 1 \\

0 &1 &0

\end{bmatrix}, C=\begin{bmatrix}

1 & 2\\

0& 6

\end{bmatrix}, \mathbf{v}=\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}.\]
Then compute and simplify the following expression.

\[\mathbf{v}^{\trans}\left( A^{\trans}-(A-B)^{\trans}\right)C.\]

Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\R^n$ such that their length are

\[||\mathbf{a}||=||\mathbf{b}||=1\]
and the inner product

\[\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.\]

Then determine the length $||\mathbf{a}-\mathbf{b}||$.

(Note that this length is the distance between $\mathbf{a}$ and $\mathbf{b}$.)

For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by

\[A=\begin{bmatrix}

\cos\theta & -\sin\theta & 0 \\

\sin\theta &\cos\theta &0 \\

0 & 0 & 1

\end{bmatrix}.\]

**(a)** Find the determinant of the matrix $A$.

**(b)** Show that $A$ is an orthogonal matrix.

**(c)** Find the eigenvalues of $A$.

Find the inverse matrix of the matrix

\[A=\begin{bmatrix}

\frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt]
\frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt]
-\frac{3}{7} & \frac{6}{7} & -\frac{2}{7}

\end{bmatrix}.\]

Let $A$ be an $n\times n$ matrix with real number entries.

Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.

Add to solve laterLet $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.

**(a) **The set $S$ consisting of all $n\times n$ symmetric matrices.

**(b)** The set $T$ consisting of all $n \times n$ skew-symmetric matrices.

**(c)** The set $U$ consisting of all $n\times n$ nonsingular matrices.

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

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 laterTest 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.

Let $A$ be an $n\times n$ matrix such that $A^k=I_n$, where $k\in \N$ and $I_n$ is the $n \times n$ identity matrix.

Show that the trace of $(A^{-1})^{\trans}$ is the conjugate of the trace of $A$. That is, show that $\tr((A^{-1})^{\trans})=\overline{\tr(A)}$.

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Find a basis for the subspace $W$ of all vectors in $\R^4$ which are perpendicular to the columns of the matrix

\[A=\begin{bmatrix}

11 & 12 & 13 & 14 \\

21 &22 & 23 & 24 \\

31 & 32 & 33 & 34 \\

41 & 42 & 43 & 44

\end{bmatrix}.\]

(Harvard University exam)

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