## 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**.

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*)

Read solution

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}.\]

In this post, we explain how to diagonalize a matrix if it is diagonalizable.

As an example, we solve the following problem.

Diagonalize the matrix

\[A=\begin{bmatrix}

4 & -3 & -3 \\

3 &-2 &-3 \\

-1 & 1 & 2

\end{bmatrix}\]
by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(Update 10/15/2017. A new example problem was added.)

Read solution

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 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*)

Consider the system of linear equations

\begin{align*}

x_1&= 2, \\

-2x_1 + x_2 &= 3, \\

5x_1-4x_2 +x_3 &= 2

\end{align*}

**(a)** Find the coefficient matrix and its inverse matrix.

**(b)** Using the inverse matrix, solve the system of linear equations.

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

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 $\tr((A^{-1})^{\trans})=\overline{\tr(A)}$.

Add to solve later

Suppose that a real matrix $A$ maps each of the following vectors

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

1 \\

1 \\

1

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

0 \\

1 \\

1

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

0 \\

0 \\

1

\end{bmatrix} \]
into the vectors

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

1 \\

2 \\

0

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

-1 \\

0 \\

3

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

3 \\

1 \\

1

\end{bmatrix},\]
respectively.

That is, $A\mathbf{x}_i=\mathbf{y}_i$ for $i=1,2,3$.

Find the matrix $A$.

(*Kyoto University Exam*)

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A square matrix $A$ is called **idempotent** if $A^2=A$.

Show that a square invertible idempotent matrix is the identity matrix.

Add to solve later