## If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$

## Problem 657

Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.

Add to solve laterof the day

Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.

Add to solve laterSuppose that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis for $\R^2$. Let $S:=[\mathbf{v}_1, \mathbf{v}_2]$.

Note that as the column vectors of $S$ are linearly independent, the matrix $S$ is invertible.

Prove that for each vector $\mathbf{v} \in V$, the vector $S^{-1}\mathbf{v}$ is the coordinate vector of $\mathbf{v}$ with respect to the basis $B$.

Add to solve laterLet $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure below.

Find the matrix representation $A$ of the linear transformation $T$.

Add to solve laterLet $A$ be a real symmetric matrix whose diagonal entries are all positive real numbers.

Is it true that the all of the diagonal entries of the inverse matrix $A^{-1}$ are also positive?

If so, prove it. Otherwise, give a counterexample.

The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.

There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).

The time limit was 55 minutes.

This post is Part 2 and contains Problem 4, 5, and 6.

Check out Part 1 and Part 3 for the rest of the exam problems.

**Problem 4**. Let

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

1 \\

2 \\

3

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

2 \\

-1 \\

4

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

0 \\

a \\

2

\end{bmatrix}.\]

Find all the values for $a$ so that the vector $\mathbf{b}$ is a linear combination of vectors $\mathbf{a}_1$ and $\mathbf{a}_2$.

**Problem 5**.

Find the inverse matrix of

\[A=\begin{bmatrix}

0 & 0 & 2 & 0 \\

0 &1 & 0 & 0 \\

1 & 0 & 0 & 0 \\

1 & 0 & 0 & 1

\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.

**Problem 6**.

Consider the system of linear equations

\begin{align*}

3x_1+2x_2&=1\\

5x_1+3x_2&=2.

\end{align*}

**(a)** Find the coefficient matrix $A$ of the system.

**(b)** Find the inverse matrix of the coefficient matrix $A$.

**(c)** Using the inverse matrix of $A$, find the solution of the system.

(*Linear Algebra Midterm Exam 1, the Ohio State University*)

Read solution

The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.

There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).

The time limit was 55 minutes.

This post is Part 1 and contains the first three problems.

Check out Part 2 and Part 3 for the rest of the exam problems.

**Problem 1.** Determine all possibilities for the number of solutions of each of the systems of linear equations described below.

**(a)** A consistent system of $5$ equations in $3$ unknowns and the rank of the system is $1$.

**(b)** A homogeneous system of $5$ equations in $4$ unknowns and it has a solution $x_1=1$, $x_2=2$, $x_3=3$, $x_4=4$.

**Problem 2.** Consider the homogeneous system of linear equations whose coefficient matrix is given by the following matrix $A$. Find the vector form for the general solution of the system.

\[A=\begin{bmatrix}

1 & 0 & -1 & -2 \\

2 &1 & -2 & -7 \\

3 & 0 & -3 & -6 \\

0 & 1 & 0 & -3

\end{bmatrix}.\]

\[A=\begin{bmatrix}

-1 & 2 & 3 & 4 & 5\\

6 & -7 & 8& 9& 10\\

11 & 12 & -13 & 14 & 15\\

16 & 17 & 18& -19 & 20\\

21 & 22 & 23 & 24 & -25

\end{bmatrix}

\] Let $I$ be the $5\times 5$ identity matrix and let $B$ be a $5\times 5$ matrix.

Suppose that $ABA^{-1}=I$.

Then determine the matrix $B$.

(*Linear Algebra Midterm Exam 1, the Ohio State University*)

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Let $I$ be the $2\times 2$ identity matrix.

Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.

An $n\times n$ matrix $A$ is called **nonsingular** if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$.

Using the definition of a nonsingular matrix, prove the following statements.

**(a)** If $A$ and $B$ are $n\times n$ nonsingular matrix, then the product $AB$ is also nonsingular.

**(b)** Let $A$ and $B$ be $n\times n$ matrices and suppose that the product $AB$ is nonsingular. Then:

- The matrix $B$ is nonsingular.
- The matrix $A$ is nonsingular. (You may use the fact that a nonsingular matrix is invertible.)

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 $A^{-1}=A$.

For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.

**(a)** $A=\begin{bmatrix}

1 & 3 & -2 \\

2 &3 &0 \\

0 & 1 & -1

\end{bmatrix}$

**(b)** $A=\begin{bmatrix}

1 & 0 & 2 \\

-1 &-3 &2 \\

3 & 6 & -2

\end{bmatrix}$.

An $n\times n$ matrix $A$ is said to be **invertible** if there exists an $n\times n$ matrix $B$ such that

- $AB=I$, and
- $BA=I$,

where $I$ is the $n\times n$ identity matrix.

If such a matrix $B$ exists, then it is known to be unique and called the **inverse matrix** of $A$, denoted by $A^{-1}$.

In this problem, we prove that if $B$ satisfies the first condition, then it automatically satisfies the second condition.

So if we know $AB=I$, then we can conclude that $B=A^{-1}$.

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

Suppose that we have $AB=I$, where $I$ is the $n \times n$ identity matrix.

Prove that $BA=I$, and hence $A^{-1}=B$.

Add to solve laterLet $A$ be an $n\times n$ nonsingular matrix with integer entries.

Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.

Add to solve laterLet $A$ be an $n\times n$ matrix.

The $(i, j)$ **cofactor** $C_{ij}$ of $A$ is defined to be

\[C_{ij}=(-1)^{ij}\det(M_{ij}),\]
where $M_{ij}$ is the $(i,j)$ **minor matrix** obtained from $A$ removing the $i$-th row and $j$-th column.

Then consider the $n\times n$ matrix $C=(C_{ij})$, and define the $n\times n$ matrix $\Adj(A)=C^{\trans}$.

The matrix $\Adj(A)$ is called the **adjoint** matrix of $A$.

When $A$ is invertible, then its inverse can be obtained by the formula

\[A^{-1}=\frac{1}{\det(A)}\Adj(A).\]

For each of the following matrices, determine whether it is invertible, and if so, then find the invertible matrix using the above formula.

**(a)** $A=\begin{bmatrix}

1 & 5 & 2 \\

0 &-1 &2 \\

0 & 0 & 1

\end{bmatrix}$.

**(b)** $B=\begin{bmatrix}

1 & 0 & 2 \\

0 &1 &4 \\

3 & 0 & 1

\end{bmatrix}$.

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 singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.

Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:

\[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\]

Using the formula, calculate the inverse matrix of $\begin{bmatrix}

2 & 1\\

1& 2

\end{bmatrix}$.

Find the inverse matrix of the $3\times 3$ matrix

\[A=\begin{bmatrix}

7 & 2 & -2 \\

-6 &-1 &2 \\

6 & 2 & -1

\end{bmatrix}\]
using the Cayley-Hamilton theorem.

10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.

The quiz is designed to test your understanding of the basic properties of these topics.

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.

Consider the following system of linear equations

\begin{align*}

2x+3y+z&=-1\\

3x+3y+z&=1\\

2x+4y+z&=-2.

\end{align*}

**(a)** Find the coefficient matrix $A$ for this system.

**(b)** Find the inverse matrix of the coefficient matrix found in (a)

**(c)** Solve the system using the inverse matrix $A^{-1}$.

Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors

\[\begin{bmatrix}

1 \\

0

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

2 \\

1

\end{bmatrix},\]
respectively.