Determinants of Matrices
Summary
-
Let $A, B$ be $n\times n$ matrices.
- $A$ is nonsingular if and only if $\det(A)\neq 0$.
- $\det(AB)=\det(A)\det(B)$.
- If $A$ is invertible, then $\det(A^{-1})=\det(A)^{-1}$.
=solution
Problems
- 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 $A$. -
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. -
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) -
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. -
Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
\[A=\begin{bmatrix}
x & x^2 & 1 \\
2 &3 &1 \\
0 & -1 & 1
\end{bmatrix}.\] -
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) -
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.) -
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. -
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 $A$. - 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 $A^{-1}$. - Given any constants $a,b,c$ where $a\neq 0$, find all values of $x$ such that the matrix $A$ is invertible if $A=
\begin{bmatrix}
1 & 0 & c \\
0 & a & -b \\
-1/a & x & x^{2}
\end{bmatrix}$. - 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) - 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?
- Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd.