Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)
Problem 136
Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.
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Add to solve later Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$
Problem 135
Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.
(a) $\rk(AB) \leq \rk(A)$.
(b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.
Add to solve later Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?
Problem 133
Find a square root of the matrix
\[A=\begin{bmatrix}
1 & 3 & -3 \\
0 &4 &5 \\
0 & 0 & 9
\end{bmatrix}.\]
How many square roots does this matrix have?
(University of California, Berkeley Qualifying Exam)
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Add to solve later Find a Basis For the Null Space of a Given $2\times 3$ Matrix
Problem 132
Let
\[A=\begin{bmatrix}
1 & 1 & 0 \\
1 &1 &0
\end{bmatrix}\]
be a matrix.
Find a basis of the null space of the matrix $A$.
(Remark: a null space is also called a kernel.)
Add to solve later Find Values of $a$ so that the Matrix is Nonsingular
Problem 126
Let $A$ be the following $3 \times 3$ matrix.
\[A=\begin{bmatrix}
1 & 1 & -1 \\
0 &1 &2 \\
1 & 1 & a
\end{bmatrix}.\]
Determine the values of $a$ so that the matrix $A$ is nonsingular.
Add to solve later The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$
Problem 121
Let $A$ be an $m \times n$ real matrix. Then the null space $\calN(A)$ of $A$ is defined by
\[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\]
That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$.
Prove that the null space $\calN(A)$ is a subspace of the vector space $\R^n$.
(Note that the null space is also called the kernel of $A$.)
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Add to solve later Express a Vector as a Linear Combination of Other Vectors
Problem 115
Express the vector $\mathbf{b}=\begin{bmatrix}
2 \\
13 \\
6
\end{bmatrix}$ as a linear combination of the vectors
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
5 \\
-1
\end{bmatrix},
\mathbf{v}_2=
\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix},
\mathbf{v}_3=
\begin{bmatrix}
1 \\
4 \\
3
\end{bmatrix}.\]
(The Ohio State University, Linear Algebra Exam)
Add to solve later Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector
Problem 114
Let
\[A=\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix}
1\\
0
\end{bmatrix}.\]
Compute $A^{2017}\mathbf{u}$.
(The Ohio State University, Linear Algebra Exam)
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Add to solve later Symmetric Matrices and the Product of Two Matrices
Problem 111
Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.
(a) The product $AB$ is symmetric if and only if $AB=BA$.
(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.
Add to solve later 10 True or False Problems about Basic Matrix Operations
Problem 104
Test your understanding of basic properties of matrix operations.
There are 10 True or False Quiz Problems.
These 10 problems are very common and essential.
So make sure to understand these and don’t lose a point if any of these is your exam problems.
(These are actual exam problems at the Ohio State University.)
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.
Add to solve later Find the Rank of a Matrix with a Parameter
Problem 103
Find the rank of the following real matrix.
\[ \begin{bmatrix}
a & 1 & 2 \\
1 &1 &1 \\
-1 & 1 & 1-a
\end{bmatrix},\]
where $a$ is a real number.
(Kyoto University, Linear Algebra Exam)
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Add to solve later Possibilities For the Number of Solutions for a Linear System
Problem 102
Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.
(a) \[\left\{
\begin{array}{c}
ax+by=c \\
dx+ey=f,
\end{array}
\right.
\]
where $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.
(b) $A \mathbf{x}=\mathbf{0}$, where $A$ is a singular matrix.
(c) A homogeneous system of $3$ equations in $4$ unknowns.
(d) $A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:
\[\begin{bmatrix}
1 & 0 & -1 & 0 \\
0 &1 & 2 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.\]
(The Ohio State University, Linear Algebra Exam)
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Add to solve later Determine When the Given Matrix Invertible
Problem 101
For which choice(s) of the constant $k$ is the following matrix invertible?
\[A=\begin{bmatrix}
1 & 1 & 1 \\
1 &2 &k \\
1 & 4 & k^2
\end{bmatrix}.\]
(Johns Hopkins University, Linear Algebra Exam)
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Add to solve later If the Matrix Product $AB=0$, then is $BA=0$ as Well?
Problem 98
Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix.
Is it true that the matrix product with opposite order $BA$ is also the zero matrix?
If so, give a proof. If not, give a counterexample.
Add to solve later True or False: $(A-B)(A+B)=A^2-B^2$ for Matrices $A$ and $B$
Problem 96
Let $A$ and $B$ be $2\times 2$ matrices.
Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2-B^2$.
Add to solve later Quiz: Linear Equations and Matrix Entreis
Companion Matrix for a Polynomial
Problem 85
Consider a polynomial
\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.
Define the matrix
\[A=\begin{bmatrix}
0 & 0 & \dots & 0 &-a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
0 & 1 & \dots & 0 & -a_2 \\
\vdots & & \ddots & & \vdots \\
0 & 0 & \dots & 1 & -a_{n-1}
\end{bmatrix}.\]
Then prove that the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$.
The matrix is called the companion matrix of the polynomial $p(x)$.
Add to solve later If a Power of a Matrix is the Identity, then the Matrix is Diagonalizable
Problem 84
Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix.
Show that the matrix $A$ is diagonalizable.
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Add to solve later The Vector Space Consisting of All Traceless Diagonal Matrices
Problem 79
Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.
That is,
\begin{equation*}
V:=\left\{ A=\begin{bmatrix}
a_{11} & 0 & \dots & 0 \\
0 &a_{22} & \dots & 0 \\
0 & 0 & \ddots & \vdots \\
0 & 0 & \dots & a_{nn}
\end{bmatrix} \quad \middle| \quad
\begin{array}{l}
a_{11}, \dots, a_{nn} \in \C,\\
\tr(A)=0 \\
\end{array}
\right\}
\end{equation*}
Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.
(a) Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)
(b) Show that matrices
\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.
(c) Find the dimension of $V$.
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Add to solve later Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix.
Problem 76
Let 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)
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