Miscellaneous Exercises

EXERCISE 1.3.6  

1. Complete the proofs of Theorems 1.2.5 and 1.2.11.
2. Let ${\mathbf x}= \begin{bmatrix}x_1\\ x_2 \end{bmatrix}, \; {\mathbf y}= \begin{bmatrix}y_1\\ y_2 \end{bmatrix}, \; A= \begin{bmatrix}1 & 0 \\ 0 & -1 \end{bmatrix}$ and $B = \begin{bmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}.$ Geometrically interpret ${\mathbf y}= A {\mathbf x}$ and ${\mathbf y}= B {\mathbf x}.$
3. Consider the two coordinate transformations
   $\;\;\begin{array}{ll} x_1 & = a_{11} y_1 + a_{12} y_2 \\ x_2 & = a_{21} y_1 + a_{22} y_2 \end{array}\;$ and $\;\;\begin{array}{ll} y_1 & = b_{11} z_1 + b_{12} z_2 \\ y_2 & = b_{21} z_1 + b_{22} z_2 \end{array}.$
   1. Compose the two transformations to express $x_1, x_2$ in terms of $z_1, z_2.$
   2. If ${\mathbf x}^t = [ x_1, \;x_2 ], \; {\mathbf y}^t = [ y_1, \;y_2 ]$ and ${\mathbf z}^t = [ z_1, \;z_2]$ then find matrices $A, B$ and $C$ such that ${\mathbf x}= A {\mathbf y}, \; {\mathbf y}= B {\mathbf z}$ and ${\mathbf x}= C {\mathbf z}.$
   3. Is $C = A B?$
4. For a square matrix $A$ of order $n,$ we define trace of $A,$ denoted by ${\mbox{tr }}(A)$ as
   $${\mbox{tr }}(A) = a_{11} + a_{22} + \cdots a_{nn}.$$
   Then for two square matrices, $A$ and $B$ of the same order, show the following:
   1. ${\mbox{tr }}(A + B) = {\mbox{ tr }}(A) + {\mbox{ tr }}(B).$
   2. ${\mbox{tr }}(A B) = {\mbox{ tr }}(B A).$
5. Show that, there do not exist matrices $A$ and $B$ such that $AB - B A = c I_n$ for any $c \neq 0.$
6. Let $A$ and $B$ be two $m \times n$ matrices and let ${\mathbf x}$ be an $n \times 1$ column vector.
   1. Prove that if $A {\mathbf x}= {\mathbf 0}$ for all ${\mathbf x},$ then $A$ is the zero matrix.
   2. Prove that if $A {\mathbf x}= B {\mathbf x}$ for all ${\mathbf x},$ then $A = B.$
7. Let $A$ be an $n \times n$ matrix such that $A B = B A$ for all $n \times n$ matrices $B.$ Show that $A = \alpha I$ for some $\alpha \in {\mathbb{R}}.$
8. Let $A = \begin{bmatrix}1 & 2 \\ 2 & 1 \\ 3 & 1 \end{bmatrix}.$ Show that there exist infinitely many matrices $B$ such that $B A = I_2.$ Also, show that there does not exist any matrix $C$ such that $A C = I_3.$
9. Compute the matrix product $A B$ using the block matrix multiplication for the matrices $A = \left[\begin{array}{cc\vert cc} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ \hline 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array} \right]$ and $B = \left[\begin{array}{cc\vert cc} 1 & 2 & 2 & 1 \\ 1 & 1 & 2 & 1 \\ \hline 1 & 1 & 1 & 1 \\ -1 & 1 & -1 & 1 \end{array} \right].$
10. Let $A = \begin{bmatrix}P&Q\\ R&S\end{bmatrix}.$ If $P, Q, R$ and $S$ are symmetric, what can you say about $A?$ Are $P, Q, R$ and $S$ symmetric, when $A$ is symmetric?
11. Let $A=[a_{ij}]$ and $B= [b_{ij}]$ be two matrices. Suppose ${\mathbf a}_1, \; {\mathbf a}_2, \; \ldots, \; {\mathbf a}_n$ are the rows of $A$ and ${\mathbf b}_1, \; {\mathbf b}_2, \; \ldots, \; {\mathbf b}_p$ are the columns of $B.$ If the product $A B$ is defined, then show that
    

    $$A B = [A {\mathbf b}_1, \; A {\mathbf b}_2, \; \ldots, \; A {\mathbf b}... ...} {\mathbf a}_1 B \\ {\mathbf a}_2 B \\ \vdots\\ {\mathbf a}_n B \end{bmatrix}.$$

    
    [That is, left multiplication by $A,$ is same as multiplying each column of $B$ by $A.$ Similarly, right multiplication by $B,$ is same as multiplying each row of $A$ by $B.$ ]