# 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 $A= \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix}$. Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic square. Compute the determinant of $A$.

2. 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.

3. 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)

4. 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.

5. Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
$A=\begin{bmatrix} x & x^2 & 1 \\ 2 &3 &1 \\ 0 & -1 & 1 \end{bmatrix}.$
6. 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)

7. 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.)

8. 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.

9. 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 $A$.

10. 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 $A^{-1}$.

11. Given any constants $a,b,c$ where $a\neq 0$, find all values of $x$ such that the matrix $A$ is invertible if $A= \begin{bmatrix} 1 & 0 & c \\ 0 & a & -b \\ -1/a & x & x^{2} \end{bmatrix}$.

12. 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)

13. 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?

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