Problem Sets: Matrix Algebra (矩阵代数)

Last Update: 2026-04-15

• Reduce each of the following matrices to row echelon form and reduced row echelon form:

• Let $A$ be a $3 \times 3$ matrix whose first row is $[a,b,c]^{T}$, where $a,b,c$ are not all zero. Let \(B = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & k \end{bmatrix}\)with a constant $k$. Given $AB = O$, find the general solution of the linear system $A\vec{x} = \vec{0}$.

• Prove that, for square matrices $A$ and $B$, $AB = BA$ if and only if $(A - B)(A + B) = A^{2} - B^{2}$.

• Suppose $A^{2}{=}I$ and $B^{2}{=}I$, show that $(A B)^{2}=I$ if and only if $A B=B A$.

• Matrix $A$ is called a symmetric matrix, if $A^{T}=A$. Let $A, B$ be symmetric matrices, prove that $AB$ is symmetric if and only if $AB = BA$.

• Prove that the product of two upper triangular $n \times n$ matrices is upper triangular.

• Find conditions on $a, b, c,$ and $d$ such that $B = \begin{bmatrix} a & b \ c & d \end{bmatrix}$ commutes with every $2 \times 2$ matrix.

• Find the matrix $X$ such that $X = AX + B$, where \(A = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ 3 & 3 \end{bmatrix}.\)

• Suppose that $A B C^{-1}=D$ with \(A = \begin{bmatrix} 1 & -2 & -1 \\ -2 & 5 & 2 \\-1 & -2 & 0\end{bmatrix}, C = \begin{bmatrix}1 & 2& 0 \\ 0 & 1 & -1 \\-1 & 1 & 0\end{bmatrix}, D = \begin{bmatrix}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{bmatrix},\) find matrix $B$?

• If $A=(B+2 I) C$, with \(A^{T}=\begin{bmatrix}1 & 3 \\ -3 & 5 \\ 0 & 1\end{bmatrix}, B=\begin{bmatrix}-1 & 0 \\ 2 & 1\end{bmatrix}\) and $I$ the identity matrix. Find matrix $C$.

• Let \(B=\begin{bmatrix} {1}&{0}&{0}\\ {0}&{1}&{-1}\\ {0}&{0}&{1}\end{bmatrix}, \text{ and } C=\begin{bmatrix}{2}&{1}&{3}\\ {0}&{2}&{1}\\ {0}&{0}&{2}\end{bmatrix}.\) Find matrix $A$ which satisfies \(A(I-C^{-1}B)^{T} C^{T}=I.\)

• For all matrices $A_{m \times k}$ and $B_{k \times n}$, show that the block matrix \(L = \begin{bmatrix} I - BA & B \\ 2A - ABA & AB - I \end{bmatrix}\) has the property $L^{2} = I$. Matrices with this property are said to be involutory, and they occur in the science of cryptography.

• Using induction, prove that for all $n \geq 1$, \((A_1 + A_2 + \cdots + A_n)^{T} = A_1^{T} + A_2^{T} + \cdots + A_n^{T}.\)

• Using induction, prove that for all $n \geq 1$, \((A_1 A_2 \cdots A_n)^{T} = A_n^{T} \cdots A_2^{T} A_1^{T}.\)

• Let $\vec{x}$ be a column vector having all its components equal to $0$ except the $i$-th component which is equal to $1$. Let $A$ be an arbitrary matrix, whose size is such that we can form the product $A\vec{x}$. What is $A\vec{x}$?

• Suppose that $A$ and $B$ are $m \times n$ matrices. If $A\vec{x} = B\vec{x}$ holds for all $n \times 1$ columns $\vec{x}$, prove that $A = B$.

• For $A = \begin{bmatrix} 1/2 & \alpha \ 0 & 1/2 \end{bmatrix}$, determine $\displaystyle \lim_{n \to \infty} A^{n}$.

• Find the inverse of the matrices \(\begin{bmatrix} 1 & 2 & 1 \\ 3 & 7 & 3 \\ 2 & 3 & 4 \end{bmatrix}, \quad \begin{bmatrix} 1 & -1 & 2 \\ 1 & 1 & -2 \\ 1 & 1 & 4 \end{bmatrix}.\)

• When possible, find the inverse of each of the following matrices. Check your answer by using matrix multiplication.

• Let $A$ be a square matrix.
  1. If $A^{2} = O$ show that $I - A$ is invertible.
  2. If $A^{3} = O$ show that $I - A$ is invertible.
  3. In general, if $A^{n} = O$ for some positive integer $n$, show that $I - A$ is invertible.
  4. Suppose that $A^{2} + 2A + I = O$. Show that $A$ is invertible.
  5. Suppose that $A^{3} - A + I = O$. Show that $A$ is invertible.

• If $A$, $B$, and $A+B$ are each nonsingular, prove that \(A(A + B)^{-1}B = B(A + B)^{-1}A = (A^{-1} + B^{-1})^{-1}.\)

• If $A$ is a square matrix such that $I - A$ is nonsingular, prove that \(A(I - A)^{-1} = (I - A)^{-1}A.\)

• If $A = [a_{ij}(t)]$ is a matrix whose entries are functions of a variable $t$, the derivative of $A$ with respect to $t$ is defined to be the matrix of derivatives. That is, \(\dfrac{d A}{dt} = \left[ \dfrac{da_{ij}}{dt} \right].\)Derive the product rule for differentiation \(\dfrac{d(AB)}{dt} = \dfrac{d A}{dt}B + A\dfrac{d B}{dt}.\)

• If $A$ and $I - A$ are invertible matrices, and the matrix $B$ satisfies $[I - (I - A)^{-1}]B = A$. Find the matrix $B-A$.

• The trace of a matrix is the sum of all the diagonal entries. Denote the trace of a matrix by $\operatorname{tr}(X)$ with $X$ being an $n\times n$ matrix, i.e. $\operatorname{tr}(X) = \sum\limits_{i=1}^n x_{ii}$. 1. Show that $\operatorname{tr}(AB) = \operatorname{tr}(BA)$, where $A$ and $B$ are both $n\times n$ matrices. 2. Can you find a matrix $X$, such that $AX - XA = I$, where $A$ and $X$ are both $n\times n$ matrices.