Tagged: nonsingular matrix

True or False Problems on Midterm Exam 1 at OSU Spring 2018

Problem 702

The following problems are True or False.

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

(a) If $AB=B$, then $B$ is the identity matrix.
(b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions.
(c) If $A$ is invertible, then $ABA^{-1}=B$.
(d) If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix.
(e) If $x_1=0, x_2=0, x_3=1$ is a solution to a homogeneous system of linear equation, then the system has infinitely many solutions.

 
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Are Coefficient Matrices of the Systems of Linear Equations Nonsingular?

Problem 669

(a) Suppose that a $3\times 3$ system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular?

(b) Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system nonsingular?

(c) Let $A$ be a $4\times 4$ matrix and let
\[\mathbf{v}=\begin{bmatrix}
1 \\
2 \\
3 \\
4
\end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix}
4 \\
3 \\
2 \\
1
\end{bmatrix}.\] Suppose that we have $A\mathbf{v}=A\mathbf{w}$. Is the matrix $A$ nonsingular?

 
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The Vector $S^{-1}\mathbf{v}$ is the Coordinate Vector of $\mathbf{v}$

Problem 632

Suppose 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 $B$.

 
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Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.

Problem 574

Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$.

(a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$.

(b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of $\R^3$.

 
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Linear Algebra Midterm 1 at the Ohio State University (3/3)

Problem 572

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 3 and contains Problem 7, 8, and 9.
Check out Part 1 and Part 2 for the rest of the exam problems.


Problem 7. Let $A=\begin{bmatrix}
-3 & -4\\
8& 9
\end{bmatrix}$ and $\mathbf{v}=\begin{bmatrix}
-1 \\
2
\end{bmatrix}$.

(a) Calculate $A\mathbf{v}$ and find the number $\lambda$ such that $A\mathbf{v}=\lambda \mathbf{v}$.

(b) Without forming $A^3$, calculate the vector $A^3\mathbf{v}$.


Problem 8. Prove that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.


Problem 9.
Determine whether each of the following sentences is true or false.

(a) There is a $3\times 3$ homogeneous system that has exactly three solutions.

(b) If $A$ and $B$ are $n\times n$ symmetric matrices, then the sum $A+B$ is also symmetric.

(c) If $n$-dimensional vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly dependent, then the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ is also linearly dependent for any $n$-dimensional vector $\mathbf{v}_4$.

(d) If the coefficient matrix of a system of linear equations is singular, then the system is inconsistent.

(e) The vectors
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}\] are linearly independent.

 
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Two Matrices are Nonsingular if and only if the Product is Nonsingular

Problem 562

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:

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

 
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A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors

Problem 561

Let $A$ be a singular $n\times n$ matrix.
Let
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
0 \\
\vdots \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1 \\
\vdots \\
0
\end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix}
0 \\
0 \\
\vdots \\
1
\end{bmatrix}\] be unit vectors in $\R^n$.

Prove that at least one of the following matrix equations
\[A\mathbf{x}=\mathbf{e}_i\] for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.

 
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The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$.

Problem 560

Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector.
Then the product $A\mathbf{b}$ is an $n$-dimensional vector.
Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$.

Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$.

 
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Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations

Problem 552

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) $A=\begin{bmatrix}
1 & 0 & 2 \\
-1 &-3 &2 \\
3 & 6 & -2
\end{bmatrix}$.

 
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