Dr. Xiaozhou Li

Dr. Xiaozhou Li

Mathematics. Strength. Long-Term Mindset. Think Hard. Train Harder. Live Long.

Problem Sets: Matrix Algebra (矩阵代数)

Published:

Last Update: 2026-04-05

  • Reduce each of the following matrices to row echelon form and reduced row echelon form:
    1. $\displaystyle\begin{bmatrix} 1 & 2 & 3 & 3 \ 2 & 4 & 6 & 9 \2 & 6 & 7 & 6 \end{bmatrix}$
    2. $\displaystyle \begin{bmatrix} 1 & 2 & 3 \ 2 & 6 & 8 \ 2 & 6 & 0 \ 1 & 2 & 5 \ 3 & 8 & 6\end{bmatrix}$
    3. $\displaystyle \begin{bmatrix} 2 & 1 & 1 & 3 & 0 & 4 & 1 \ 4 & 2 & 4 & 4 & 1 & 5 & 5 \2 & 1 & 3 & 1 & 0 & 4 & 3 \ 6 & 3 & 4 & 8 & 1 & 9 & 5 \ 0 & 0 & 3 & -3 & 0 & 0 & 3 \ 8 & 4 & 2 & 14 & 1 & 13 & 3 \end{bmatrix}$

  • 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 $(A - B)(A + B) = A^{2} - B^{2}$.

  • 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 \(A = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ 3 & 3 \end{bmatrix}.\)

  • 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.
    1. $\displaystyle \begin{bmatrix} 1 & 2 \1 & 3 \end{bmatrix}$
    2. $\displaystyle \begin{bmatrix} 1 & 2 \2 & 4 \end{bmatrix}$
    3. $\displaystyle \begin{bmatrix} 4 & -8 & 5 \ 4 & -7 & 4 \ 3 & -4 & 2 \end{bmatrix}$
    4. $\displaystyle \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$
    5. $\displaystyle \begin{bmatrix} 1 & 1 & 1 & 1 \ 1 & 2 & 2 & 2 \ 1 & 2 & 3 & 3 \ 1 & 2 & 3 & 4 \end{bmatrix}$
  • 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 \(\dfrac{d(AB)}{dt} = \dfrac{d A}{dt}B + A\dfrac{d B}{dt}.\)