Tagged: elementary row operations
Determine whether the Given 3 by 3 Matrices are Nonsingular
Determine whether the following matrices are nonsingular or not.
(a) $A=\begin{bmatrix}
1 & 0 & 1 \\
2 &1 &2 \\
1 & 0 & -1
\end{bmatrix}$.
(b) $B=\begin{bmatrix}
4 & 1 & 4
For What Values of $a$, Is the Matrix Nonsingular?
Determine the values of a real number $a$ such that the matrix
\[A=\begin{bmatrix}
3 & 0 & a \\
0 & 18a & a+1
\end{bmatrix}\] is nonsingular.
If Two Matrices Have the Same Rank, Are They Row-Equivalent?
If $A, B$ have the same rank, can we conclude that they are row-equivalent?
If so, then prove it. If not, then provide a counterexample.
Find a Row-Equivalent Matrix which is in Reduced Row Echelon Form and Determine the Rank
For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.
(a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$.
(b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$.
(c) $C = \begin{bmatrix} 2 & -2 & 4 \\ 4 & 1 & -2 \\ 6 & -1 & 2 \end{bmatrix}$.
(d) $D = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}$.
(e) $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$.
Row Equivalence of Matrices is Transitive
If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$?
An Example of Matrices $A$, $B$ such that $\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B)$
For an $m\times n$ matrix $A$, we denote by $\mathrm{rref}(A)$ the matrix in reduced row echelon form that is row equivalent to $A$.
For example, consider the matrix $A=\begin{bmatrix}
0 &2 &2
\end{bmatrix}$
Then we have
\end{bmatrix}
\xrightarrow{\frac{1}{2}R_2}
\begin{bmatrix}
0 &1 & 1
\xrightarrow{R_1-R_2}
\end{bmatrix}\] and the last matrix is in reduced row echelon form.
Hence $\mathrm{rref}(A)=\begin{bmatrix}
Find an example of matrices $A$ and $B$ such that
\[\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B).\]
Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
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.
1 & 3 & -2 \\
(b) $A=\begin{bmatrix}
-1 &-3 &2 \\
Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix
Consider the following system of linear equations
2x+3y+z&=-1\\
3x+3y+z&=1\\
2x+4y+z&=-2.
(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 $A^{-1}$.
Find All the Eigenvalues and Eigenvectors of the 6 by 6 Matrix
Find all the eigenvalues and eigenvectors of the matrix
10001 & 3 & 5 & 7 &9 & 11 \\
1 & 10003 & 5 & 7 & 9 & 11 \\
1 & 3 & 10005 & 7 & 9 & 11 \\
1 & 3 & 5 & 10007 & 9 & 11 \\
1 &3 & 5 & 7 & 10009 & 11 \\
1 &3 & 5 & 7 & 9 & 10011
\end{bmatrix}.\]
(MIT, Linear Algebra Homework Problem)
Determine linear transformation using matrix representation
Let $T$ be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations.
T\left(\, \begin{bmatrix}
1 \\
\end{bmatrix} \,\right)
=\begin{bmatrix}
\end{bmatrix}, \qquad T\left(\, \begin{bmatrix}
\end{bmatrix} \, \right) =
\end{bmatrix}, \qquad
T \left( \, \begin{bmatrix}
\end{bmatrix} \, \right)=
\end{bmatrix}.
Then for any vector
\[\mathbf{x}=\begin{bmatrix}
x \\
y \\
\end{bmatrix}\in \R^3,\] find the formula for $T(\mathbf{x})$.
Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix
(a) Let $A=\begin{bmatrix}
1 & 3 & 0 & 0 \\
1 &3 & 1 & 2 \\
1 & 3 & 1 & 2
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.
(b) Find the rank and nullity of the matrix $A$ in part (a).
Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space
\end{bmatrix}$ and let
\[\mathbf{a}=\begin{bmatrix}
-3 \\
\end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix}
\end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix}
\end{bmatrix}.\] For each of the vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, determine whether the vector is in the null space $\calN(A)$. Do the same for the range $\calR(A)$.
(b) Find a basis of the null space of the matrix $B=\begin{bmatrix}
-2 &-2 &-4
Given a Spanning Set of the Null Space of a Matrix, Find the Rank
Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors
\[\begin{bmatrix}
\end{bmatrix}, \begin{bmatrix}
\end{bmatrix}, \text{ and } \begin{bmatrix}
\end{bmatrix}.\] Then find the rank of the matrix $A$.
(Purdue University, Linear Algebra Final Exam Problem)
Find the Inverse Matrix of a $3\times 3$ Matrix if Exists
Find the inverse matrix of
\end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(The Ohio State University, Linear Algebra Midterm Exam Problem)
Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation
(a) Find the inverse matrix of
(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\] where
\[B=\begin{bmatrix}
4 & 1\\
2& 6
\end{bmatrix}.\] Verify that the matrix $A$ you obtained is actually a nonsingular matrix.
Find a Nonsingular Matrix Satisfying Some Relation
Determine whether there exists a nonsingular matrix $A$ if
\[A^2=AB+2A,\] where $B$ is the following matrix.
If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.
(a) \[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
\end{bmatrix}\]
(b) \[B=\begin{bmatrix}
Determine Linearly Independent or Linearly Dependent. Express as a Linear Combination
Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent, express one vector in the set as a linear combination of the others.
\[\left\{\, \begin{bmatrix}
\end{bmatrix},
\end{bmatrix}\, \right\}.\]
Linear Transformation, Basis For the Range, Rank, and Nullity, Not Injective
Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients.
Let $T: P_3 \to V$ be the linear transformation defined by
\[T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix}
a_0+a_2 & -a_0+a_3\\
a_1-a_2 & -a_1-a_3
\end{bmatrix}\] for any polynomial $a_0+a_1x+a_2x^2+a_3 \in P_3$.
Find a basis for the range of $T$, $\calR(T)$, and determine the rank of $T$, $\rk(T)$, and the nullity of $T$, $\nullity(T)$.
Also, prove that $T$ is not injective.
The Inverse Matrix of an Upper Triangular Matrix with Variables
Let $A$ be the following $3\times 3$ upper triangular matrix.
1 & x & y \\
0 &1 &z \\
\end{bmatrix},\] where $x, y, z$ are some real numbers.
Determine whether the matrix $A$ is invertible or not. If it is invertible, then find the inverse matrix $A^{-1}$.
Find All Matrices $B$ that Commutes With a Given Matrix $A$: $AB=BA$
\end{bmatrix}.\] Then
(a) Find all matrices
x & y\\
z& w
\end{bmatrix}\] such that $AB=BA$.
(b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that
\[AB=BA \text{ and } AC \neq CA.\]
Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems
Using Properties of Inverse Matrices, Simplify the Expression
True or False Problems on Midterm Exam 1 at OSU Spring 2018
Determine Whether Given Matrices are Similar