2024春 线性代数期末(回忆版)

1 Multiple Choice: Only one choice is correct

1-(1)

Let $A$ be an $m\times m$ real symmetric matrix. Which of the following assertions is false？()

(A) If $v_1$ and $v_2$ are two m-dimensional real column vectors which satisfy $A{v}_1=v_1$ and $A{v}_2=0$, then $v_1$ and $v_2$ are orthogonal

(B) There exists a real invertible matrix $P$ such that $P^{-1}AP$ is diagonal

(C) A has $m$ distinct eigenvalues

(D) The sum of algebraic multiplicities of the distinct eigenvalues of $A$ is $m$

1-(2)

Let $A$ be an $m\times n$ real matrix and $U\mathrm{\Sigma }{V}^{\mathrm{\top }}$ be a singular value decomposition of $A$. Which of the following assertions is false？()

(A) The columns of $U$ are eigenvectors of $A{A}^{\mathrm{\top }}$

(B) The columns of $V$ are eigenvectors of $A^{\mathrm{\top }}A$

(C) The eigenvalues of $A{A}^{\mathrm{\top }}$ and $A^{\mathrm{\top }}A$ are real and positive

(D) $A{A}^{\mathrm{\top }}$ and $A^{\mathrm{\top }}A$ have the same set of positive eigenvalues

1-(3)

Let $A$ be an $n\times n$ real matrix. If for any $x\in R^n$, we have $x^{\mathrm{\top }}Ax=0$, then( )

(A) $|A|=0$

(B) $A^{\mathrm{\top }}=-A$

(C) $A=0$. Where 0 denotes the $n\times n$ zero matrix

(D) the eigenvalues of $A$ are all zero

1-(4)

Which of the following matrices is NOT diagonalizable？( )

(A) $\left[\begin{array}{lll}0& 0& 0\\ 1& 0& 0\\ 0& 2& 1\end{array}\right]$

(B) $\left[\begin{array}{ccc}1& 2& -1\\ 2& 0& 0\\ -1& 0& 0\end{array}\right]$

(C) $\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 2& -1\end{array}\right]$

(D) $\left[\begin{array}{lll}0& 0& 0\\ 0& 1& 0\\ 0& 1& 2\end{array}\right]$

1-(5)

Let $A$ be a $3\times 3$ permutation matrix. Then $\det (-{\left(A^{\mathrm{\top }}\right)}^{2024})$ equals( )

(A) 0

(B) 2024

(C) 1

(D) -1

2 Fill in the blanks

2-(1)

Suppose $A$ =

$$\left[\begin{array}{ccc}a& b& -\frac{3}{7}\\ -\frac{3}{7}& c& \frac{2}{7}\\ \frac{2}{7}& d& e\end{array}\right]$$

is an orthogonal matrix. Then $a=$ $\underset{―}{}$, $e=$ $\underset{―}{}$

2-(2)

Let $A$ and $B$ be two $n\times n$ real matrices $|A|=4,|B|=3$ and $|A^{-1}+B|=2$, then $|A+B^{-1}|=$ $\underset{―}{}$

2-(3)

Let $A$ be a $3\times 3$ matrix. Suppose $|A|=-5$ and $A^2+4A-5I=0$, then the three eigenvalues of $A$ are $\underset{―}{}$. Where I denotes the $3\times 3$ identity matrix and

0 denotes the $3\times 3$ zero matrix

2-(4)

Suppose

$$A=\left[\begin{array}{lll}1& a& 1\\ a& b& a\\ 1& a& 1\end{array}\right]\text{ and }B=\left[\begin{array}{lll}2& 0& 0\\ 0& b& 0\\ 0& 0& 0\end{array}\right]$$

are similar, then $a=$ $\underset{―}{}$

3

Consider the following matrix $A=\left[\begin{array}{lll}4& 2& 2\\ 2& 4& 2\\ 2& 2& 4\end{array}\right]$

3-(a)

Find all the eigenvalues of $A$

3-(b)

Find an orthogonal matrix $Q$ such that $Q^{-1}AQ$ is diagonal

3-(c)

Compute $A^k$ for any positive integer $K$

4

Find the determinant of the following $7\times 7$ matrix:

$$A=\left[\begin{array}{lllllll}5& 3& 0& 0& 0& 0& 0\\ 2& 5& 3& 0& 0& 0& 0\\ 0& 2& 5& 3& 0& 0& 0\\ 0& 0& 2& 5& 3& 0& 0\\ 0& 0& 0& 2& 5& 3& 0\\ 0& 0& 0& 0& 2& 5& 3\\ 0& 0& 0& 0& 0& 2& 5\end{array}\right]$$

5

Let $A=\left[\begin{array}{ll}1& 1\\ 2& 3\\ 4& 5\end{array}\right]$

5-(a)

Find all the eigenvalues and their corresponding linearly independent eigenvectors of $P=A{\left(A^{\mathrm{\top }}A\right)}^{-1}{A}^{\mathrm{\top }}$

5-(b)

Show that $P$ is diagonalizable

6

Consider the following quadratic form:

$$Q(x,y,z)=\lambda (x^2+y^2+z^2)+2xy+2xz-2yz$$

6-(a)

For what values of $\lambda$ is $Q(x,y,z)$ positive definite?

6-(b)

For what values of $\lambda$ is $Q(x,y,z)$ negative definite?

6-(c)

Find the type of quadric surface defined by the following equation:

$$3(x^2+y^2+z^2)+2xy+2xz-2yz=1$$

7

Let $A$ and $B$ be $n\times n$ real matrices. The trace of $A$ is defined to be the sum of all of its diagonal entries:

$$\mathrm{tr}(A)=a_{11}+a_{22}+\cdots +a_{nn}$$

7-(a)

Suppose $A$ is similar to $B$, prove that

$$\mathrm{tr}(A)=\mathrm{tr}(B)$$

7-(b)

Let $A$ and $B$ be real symmetric positive semidefinite matrices. Show that $\mathrm{tr}(AB)>0$

7-(c)

Suppose $A$ and $B$ are real symmetric positive semidefinite matrices and $\mathrm{tr}(AB)=0$. show that $AB=0$. Where 0 denotes the $n\times n$ zero matrix