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)

An $n\times n$ matrix is invertible if and only if its rank is $n$.
The rank of a matrix is the number of nonzero rows of a (reduced) row echelon form matrix that is row equivalent to the given matrix.

Solution.

We compute the rank of the matrix $A$.
Applying elementary row operations, we obtain
\begin{align*}
\begin{bmatrix}
1 & 1 & 1 \\
1 &2 &k \\
1 & 4 & k^2
\end{bmatrix}
\xrightarrow{\substack{R_2-R_1 \\ R_3-R_1}}
\begin{bmatrix}
1 & 1 & 1 \\
0 & 1 &k-1 \\
0 & 3 & k^2-1
\end{bmatrix}
\xrightarrow{\substack{R_1-R_2 \\ R_3-3R_2}}
\begin{bmatrix}
1 & 0 & 2-k \\
0 &1 &k-1 \\
0 & 0 & k^2-3k+2
\end{bmatrix}.
\end{align*}
The last matrix is in row echelon form.

Note that $A$ is an invertible matrix if and only if its rank is $3$.
Therefore the $(3,3)$-entry of the last matrix must be nonzero: $k^2-3k+2=(k-1)(k-2)\neq 0$.

It follows that the matrix $A$ is invertible for any $k$ except $k=1, 2$.

Projection to the subspace spanned by a vector
Let $T: \R^3 \to \R^3$ be the linear transformation given by orthogonal projection to the line spanned by $\begin{bmatrix}
1 \\
2 \\
2
\end{bmatrix}$.
(a) Find a formula for $T(\mathbf{x})$ for $\mathbf{x}\in \R^3$.
(b) Find a basis for the image subspace of $T$.
(c) Find […]

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 […]

Given the Characteristic Polynomial, Find the Rank of the Matrix
Let $A$ be a square matrix and its characteristic polynomial is give by
\[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\]
Find the rank of $A$.
(The Ohio State University, Linear Algebra Final Exam Problem)
Solution.
Note that the degree of the characteristic polynomial […]

Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix.
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 […]

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 […]