Determinants of Matrices

Determinants of Matrices

Summary

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

  1. $A$ is nonsingular if and only if $\det(A)\neq 0$.
  2. $\det(AB)=\det(A)\det(B)$.
  3. If $A$ is invertible, then $\det(A^{-1})=\det(A)^{-1}$.

=solution

Problems

  1. Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$. Is it true that
    \[\det(A+B)=\det(A)+\det(B)?\] If so, then give a proof. If not, then give a counterexample.

  2. Let
    \[ A=\begin{bmatrix}
    2 & 0 & 10 \\
    0 &7+x &-3 \\
    0 & 4 & x
    \end{bmatrix}.\] Find all values of $x$ such that $A$ is invertible.
    (Stanford University)

  3. Let
    \[A=\begin{bmatrix}
    1 & -x & 0 & 0 \\
    0 &1 & -x & 0 \\
    0 & 0 & 1 & -x \\
    0 & 1 & 0 & -1
    \end{bmatrix}\] be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.

  4. Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
    \[A=\begin{bmatrix}
    x & x^2 & 1 \\
    2 &3 &1 \\
    0 & -1 & 1
    \end{bmatrix}.\]
  5. Find the value(s) of $h$ for which the following set of vectors
    \[\left \{ \mathbf{v}_1=\begin{bmatrix}
    1 \\
    0 \\
    0
    \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
    h \\
    1 \\
    -h
    \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
    1 \\
    2h \\
    3h+1
    \end{bmatrix}\right\}\] is linearly independent.
    (Boston College)

  6. Let
    \[A=\begin{bmatrix}
    1 & 0 & 3 \\
    4 &5 &6 \\
    7 & 0 & 9
    \end{bmatrix} \text{ and } B=\begin{bmatrix}
    2 & 0 & 0 \\
    0 & 3 &0 \\
    0 & 0 & 4
    \end{bmatrix}.\] Then find the value of
    \[\det(A^2B^{-1}A^{-2}B^2).\] (Without a proof, you may assume that $A$ and $B$ are invertible matrices.)

  7. Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.
    Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity matrix.

  8. Let $A$ be a $3 \times 3$ matrix. Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have
    \[A\mathbf{x}=\begin{bmatrix}
    1 \\
    0 \\
    1
    \end{bmatrix}, A\mathbf{y}=\begin{bmatrix}
    0 \\
    1 \\
    0
    \end{bmatrix}, A\mathbf{z}=\begin{bmatrix}
    1 \\
    1 \\
    1
    \end{bmatrix}.\] Then find the value of the determinant of the matrix $A$.

  9. Determine the values of $x$ so that the matrix
    \[A=\begin{bmatrix}
    1 & 1 & x \\
    1 &x &x \\
    x & x & x
    \end{bmatrix}\] is invertible. For those values of $x$, find the inverse matrix $A^{-1}$.

  10. Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix.
    (The Ohio State University)

  11. Let $n$ be an odd positive integer. Determine whether there exists an $n \times n$ real matrix $A$ such that $A^2+I=O$, where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix. If such a matrix $A$ exists, find an example. If not, prove that there is no such $A$. How about when $n$ is an even positive number?

  12. Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd.