(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
First we look at the rank 1 case.
For a $3 \times 3$ matrix in reduced row echelon form to have rank 1, it must have 2 rows which are all 0s. The non-zero row must be the first row, and it must have a leading 1.
These conditions imply that the matrix must be of one of the following forms:
\[\begin{bmatrix} 1 & a & b \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , \quad \begin{bmatrix} 0 & 1 & c \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , \mbox{ or } \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}.\]
(b) Find all such matrices with rank 2.
For a rank 2 $3 \times 3$ matrix in reduced row echelon form, there must be one row, the bottom one, which has only 0s.
Thus we need two leading 1s in distinct columns, and every other term in the same column with a leading 1 must be 0. The possibilities are:
\[\begin{bmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 0 \end{bmatrix} , \quad \begin{bmatrix} 1 & a & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} , \mbox{ or } \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}.\]
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 […]
Find the Rank of a Matrix with a Parameter
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)
Solution.
The rank is the number of nonzero rows of a […]
Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$
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 $\rk(AB)=\rk(A)$.
Hint.
The rank of an $m \times n$ matrix $M$ is the dimension of the range […]
If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal?
Recall that a matrix $A$ is symmetric if $A^\trans = A$, where $A^\trans$ is the transpose of $A$.
Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it.
Otherwise, provide a counterexample.
Proof. […]
Linear Transformation to 1-Dimensional Vector Space and Its Kernel
Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.
Prove the followings.
(a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.
(b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the […]
Determine When the Given Matrix Invertible
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)
Hint.
An $n\times n$ matrix is […]
Rank and Nullity of a Matrix, Nullity of Transpose
Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$.
The dimension of the nullspace of $A$ is called the nullity of $A$.
Prove the followings.
(a) $\calN(A)=\calN(A^{\trans}A)$.
(b) $\rk(A)=\rk(A^{\trans}A)$.
Hint.
For part (b), […]
Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix
(a) Let $A=\begin{bmatrix}
1 & 3 & 0 & 0 \\
1 &3 & 1 & 2 \\
1 & 3 & 1 & 2
\end{bmatrix}$.
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).
Solution.
(a) […]