Problem Sets: Determinants (行列式)

Last Update: 2026-04-09

• How many multiplications are required to calculate an $n\times n$ determinant by cofactor expansion? What does the result tell you?

• Let $A$ be an $n\times n$ matrix such that $A^{2}=I$. Show that $\det A=\pm 1$.

• Show that If $A$ is an $n\times n$ matrix, then $\det (kA)=k^{n}\det A$.

• Show that if $A$ is invertible, then $\det A^{-1} = \dfrac{1}{\det A}$

• Let $A$ be an $n\times n$ matrix, show that $A \operatorname{adj}A = \det A \cdot I$.

• Compute

• Compute

• Compute \(\begin{vmatrix} a_1+b & a_2 & a_3 & \cdots & a_n \\ a_1 & a_2+b & a_3 & \cdots & a_n \\ a_1 & a_2 & a_3+b & \cdots & a_n \\ \vdots & \vdots & \vdots & \ddots & \vdots\\ a_1 & a_2 & a_3 & \cdots & a_n+b \end{vmatrix}, \quad \begin{vmatrix} a_1+b_1 & a_2 & a_3 & \cdots & a_n \\ a_1 & a_2+b_2 & a_3 & \cdots & a_n \\ a_1 & a_2 & a_3+b_3 & \cdots & a_n \\ \vdots & \vdots & \vdots & \ddots & \vdots\\ a_1 & a_2 & a_3 & \cdots & a_n+b_n \end{vmatrix}\)

• Compute the determinant of the following matrix. (Hint: is this matrix invertible?)

• Suppose \(A_1 = \begin{bmatrix} B & O \\D & C\end{bmatrix}, \,\, A_2 = \begin{bmatrix} B & D \\O & C\end{bmatrix}, \,\, A_3 = \begin{bmatrix} O & B \\C & D\end{bmatrix}, \,\, A_4 = \begin{bmatrix} D & B \\C & O\end{bmatrix},\) where $B$ and $C$ are invertible, find their determinants.

• Suppose $A$ is $n\times n$ matrix, and $C$ is $m\times m$ matrix. If $\lvert A\rvert \neq 0, \lvert D - C A^{-1} B\rvert \neq 0$, compute $\begin{bmatrix} A & B \ C & D\end{bmatrix}^{-1}$.

• For what values of $a, b$ the homogeneous linear system \(\begin{cases} ax_1 + bx_2 + bx_3 + \cdots + bx_n = 0, \\ bx_1 + ax_2 + bx_3 + \cdots + bx_n = 0, \\ \cdots\cdots \\ bx_1 + bx_2 + bx_3 + \dots + ax_n = 0 \end{cases}\) has only the trivial solution, infinitely many solutions. When there are infinitely many solutions, find all solutions.

• Given the system of linear equations \(\begin{cases} ax_1 + x_3 = 1, \\ x_1 + ax_2 + x_3 = 0, \\ x_1 + 2x_2 + ax_3 = 0, \\ ax_1 + bx_2 = 2 \end{cases}\) which has a solution, where $a$ and $b$ are constants. If \(\begin{vmatrix} a & 0 & 1 \\ 1 & a & 1 \\ 1 & 2 & a \end{vmatrix} = 4,\) find the value of \(\begin{vmatrix} 1 & a & 1 \\ 1 & 2 & a \\a & b & 0 \end{vmatrix}\) (The result should not contain $a$ or $b$).

• Consider the system of linear equations \(\begin{cases} x_1 + a_1 x_2 + a_1^2 x_3 = a_1^3, \\ x_1 + a_2 x_2 + a_2^2 x_3 = a_2^3, \\ x_1 + a_3 x_2 + a_3^2 x_3 = a_3^3, \\ x_1 + a_4 x_2 + a_4^2 x_3 = a_4^3. \end{cases}\)
  1. Prove that if $a_1, a_2, a_3, a_4$ are distinct, then the system has no solution.
  2. Let $a_1 = a_3 = k$, $a_2 = a_4 = -k$ with $k \neq 0$. Assume that $\vec{\beta}_1$ and $\vec{\beta}_2$ are two solutions of the system, where \(\vec{\beta}_1 = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}, \quad \vec{\beta}_2 = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}.\) Find the general solution of this system.
• For the $n \times n$ matrix

\(A = \begin{pmatrix} 0 & 0 & 0 & \dots & 0 & a_0 \\ -1 & 0 & 0 & \dots & 0 & a_1 \\ 0 & -1 & 0 & \dots & 0 & a_2 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 0 & a_{n-2} \\ 0 & 0 & 0 & \dots & -1 & a_{n-1} \end{pmatrix}\) compute $\det(A + tI)$, where $I$ is $n \times n$ identity matrix. You should get a nice expression involving $a_0, a_1, \dots, a_{n-1}$ and $t$. Row expansion and induction is probably the best way to go.

• Let $a_1, a_2, \dots, a_n$ be $n$ real numbers. Prove that \(\begin{vmatrix} 1 & a_1 & a_1^2 & \dots & a_1^{n-1} \\ 1 & a_2 & a_2^2 & \dots & a_2^{n-1} \\ 1 & a_3 & a_3^2 & \dots & a_3^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & a_n & a_n^2 & \dots & a_n^{n-1} \end{vmatrix} = \prod_{1 \le i < j \le n} (a_j - a_i)\) where $\prod_{1 \le i < j \le n} (a_j - a_i)$ means the product of all terms of the form $(a_j - a_i)$, where $i < j$ and both $i$ and $j$ are between $1$ and $n$. [The determinant of a matrix of this form (or its transpose) is called a Vandermonde determinant, named after the French mathematician A. T. Vandermonde (1735–1796).]

• Let $A$ be an $n\times n$ invertible matrix, show that $\operatorname{adj}(\operatorname{adj}A)=(\det A)^{n-2}A$.

• Let $A$ be an $n\times n$ matrix, show that $\operatorname{adj}(\operatorname{adj}A)=(\det A)^{n-2}A$.

• Let $A$ be a $4\times 4$ matrix, and $\det A=-2$, calculate: $\det A^{3}$, $\det 2A$, $\det A^{T}A$, $\det A A^{T}$, $\det(-A)$, $\det A^{-1}$ and $\det(\operatorname{adj}A)$.

• Let $A$ be a $3\times 3$ matrix, and $\det A = 3$, calculate $\det(A - (2\operatorname{adj}A)^{-1})$.

• Suppose matrices $A$, $B$ are $n\times n$ matrices, and $A$ and $I- AB$ are invertible, show that $I - BA$ is invertible.

• Let $A$ be an $n\times n$ matrix, and $A A^{T}=I$, $\det A=-1$, show that $\det(I+A)=0$.

• Use Cramer’s rule to compute the solutions of the systems, or find $A^{-1}$.