## The Inverse Matrix is Unique

## Problem 251

Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.

Add to solve laterLet $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.

Add to solve later Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies

\[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\]
Define the matrix

\[A=I+\mathbf{u}\mathbf{v}^{\trans}.\]

Prove that $A$ is invertible and the inverse matrix is given by the formula

\[A^{-1}=I-a\mathbf{u}\mathbf{v}^{\trans},\]
where

\[a=\frac{1}{1+\mathbf{v}^{\trans}\mathbf{u}}.\]
This formula is called the **Sherman-Woodberry formula**.

Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations.

\[A= \left[\begin{array}{rrr|r}

1 & 2 & 3 & 4 \\

2 &-1 & -2 & a^2 \\

-1 & -7 & -11 & a

\end{array} \right],\]
where $a$ is a real number. Determine all the values of $a$ so that the corresponding system is consistent.

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 $d-cb \neq 0$.

Read solution

Let

\[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 $B$.

Add to solve laterLet $A$ be an $n \times n$ matrix. Suppose that all the eigenvalues $\lambda$ of $A$ are real and satisfy $\lambda <1$.

Then show that the determinant \[ \det(I-A) >0,\] where $I$ is the $n \times n$ identity matrix.

Add to solve laterLet $V$ denote the vector space of all real $n\times n$ matrices, where $n$ is a positive integer.

Determine whether the set $U$ of all $n\times n$ nilpotent matrices is a subspace of the vector space $V$ or not.

Add to solve laterSuppose that $n\times n$ matrices $A$ and $B$ are similar.

Then show that the nullity of $A$ is equal to the nullity of $B$.

In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of $B$.

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 $A$.

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 $C$.

Read solution Add to solve later

Let

\[A=\begin{bmatrix}

1 & 3 & 3 \\

-3 &-5 &-3 \\

3 & 3 & 1

\end{bmatrix} \text{ and } B=\begin{bmatrix}

2 & 4 & 3 \\

-4 &-6 &-3 \\

3 & 3 & 1

\end{bmatrix}.\]
For this problem, you may use the fact that both matrices have the same characteristic polynomial:

\[p_A(\lambda)=p_B(\lambda)=-(\lambda-1)(\lambda+2)^2.\]

**(a)** Find all eigenvectors of $A$.

**(b)** Find all eigenvectors of $B$.

**(c)** Which matrix $A$ or $B$ is diagonalizable?

**(d)** Diagonalize the matrix stated in (c), i.e., find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$ or $B=PDP^{-1}$.

(*Stanford University Linear Algebra Final Exam Problem*)

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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 $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.

Add to solve laterLet $A$ 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 laterShow that eigenvalues of a Hermitian matrix $A$ are real numbers.

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

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Let

\[ A=\begin{bmatrix}

5 & 2 & -1 \\

2 &2 &2 \\

-1 & 2 & 5

\end{bmatrix}.\]

Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.

Your score of this problem is equal to that dimension times five.

(*The Ohio State University Linear Algebra Practice Problem*)

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Find the value(s) of $h$ for which the following set of vectors

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

1 \\

0 \\

0

\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}

h \\

1 \\

-h

\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}

1 \\

2h \\

3h+1

\end{bmatrix}\right\}\]
is linearly independent.

(*Boston College, Linear Algebra Midterm Exam Sample Problem*)

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Let $A$ be a $3 \times 3$ matrix.

Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have

\[A\mathbf{x}=\begin{bmatrix}

1 \\

0 \\

1

\end{bmatrix}, A\mathbf{y}=\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}, A\mathbf{z}=\begin{bmatrix}

1 \\

1 \\

1

\end{bmatrix}.\]

Then find the value of the determinant of the matrix $A$.

Add to solve laterLet

\[A=\begin{bmatrix}

1 & -1\\

2& 3

\end{bmatrix}.\]

Find the eigenvalues and the eigenvectors of the matrix

\[B=A^4-3A^3+3A^2-2A+8E.\]

(*Nagoya University Linear Algebra Exam Problem*)

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Prove that the matrix

\[A=\begin{bmatrix}

1 & 1.00001 & 1 \\

1.00001 &1 &1.00001 \\

1 & 1.00001 & 1

\end{bmatrix}\]
has one positive eigenvalue and one negative eigenvalue.

(*University of California, Berkeley Qualifying Exam Problem*)

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