Miscellaneous Exercises

EXERCISE 2.7.1  

1. Let $A$ be an orthogonal matrix. Show that $\det A = \pm 1.$
2. If $A$ and $B$ are two $n \times n$ non-singular matrices, are the matrices $A + B$ and $\;A - B$ non-singular? Justify your answer.
3. For an $n \times n$ matrix $A,$ prove that the following conditions are equivalent:
   1. $A$ is singular ($A^{-1}$ doesn't exist).
   2. rank $(A) \neq n.$
   3. $\det (A) = 0.$
   4. $A$ is not row-equivalent to $I_n,$ the identity matrix of order $n.$
   5. $A {\mathbf x}= {\mathbf 0}$ has a non-trivial solution for some ${\mathbf x}$ (that is, there exist $x_1, x_2, \ldots, x_n$ not all zero, such that $A \; [x_1 \; x_2 \; \cdots \; x_n]^t = {\mathbf 0}$ ).
   6. $A {\mathbf x}= b$ doesn't have a unique solution, i.e., it has no solutions or it has infinitely many solutions.
4. Let $A = \begin{bmatrix}2 & 0 & 6 & 0 & 4 \\ 5 & 3 & 2 & 2 & 7 \\ 2 & 5 & 7 & 5 & 5 \\ 2 & 0 & 9 & 2 & 7 \\ 7 & 8 & 4 & 2 & 1 \end{bmatrix}.$ We know that the numbers $20604, 53227,$ $25755, 20927$ and $78421$ are all divisible by $17.$ Does this imply $17$ divides $\det (A)$ ?
5. Let $A = [a_{ij}]_{n \times n}$ where $a_{ij} = x^{j-1}_{i}.$ Show that $\;\; \det (A) = \prod\limits_{1\le i < j\le n} (x_{j} - x_{i}).$ [The matrix $A$ is usually called the Van-dermonde matrix.]
6. Let $A=[a_{ij}]$ with $a_{ij} = \max\{i,j\}$ be an $n \times n$ matrix. Compute $\det A.$
7. Let $A=[a_{ij}]$ with $a_{ij} = {1 / {(i+j)}}$ be an $n \times n$ matrix. Show that $A$ is invertible.
8. Solve the following system of equations by Cramer's rule.
   $i) \; x + y + z - w = 1, \; x + y - z + w = 2,$ $\;2 x + y + z - w = 7, \; x + y + z + w = 3.$
   $ii) \; x - y + z - w = 1, \; x + y - z + w = 2,$ $\; 2 x + y - z - w = 7, \; x - y - z + w = 3.$
9. Suppose $A=[a_{ij}]$ and $B= [b_{ij}]$ are two $n \times n$ matrices such that $b_{ij} = p^{i-j} a_{ij}$ for $1 \le i, j \le n$ for some non-zero real number $p$ . Then compute $\det(B)$ in terms of $\det (A)$ .
10. The position of an element $a_{ij}$ of a determinant is called even or odd according as $i + j$ is even or odd. Show that
    1. If all the entries in odd positions are multiplied with $-1$ then the value of the determinant doesn't change.
    2. If all entries in even positions are multiplied with $-1$ then the determinant
       1. does not change if the matrix is of even order.
       2. is multiplied by $-1$ if the matrix is of odd order.
11. Let $A$ be an $n \times n$ Hermitian matrix, that is, $A^{*} = A.$ Show that $\det A$ is a real number. [$A$ is a matrix with complex entries and $A^{*} = \overline{A^{t}}.$ ]
12. Let $A$ be an $n \times n$ matrix. Then show that
    $$A {\mbox{ is invertible}} \Longleftrightarrow Adj (A) {\mbox{ is invertible}}.$$

13. Let $A$ and $B$ be invertible matrices. Prove that ${\mbox{Adj}}(AB) = {\mbox{Adj}}(B) {\mbox{Adj}}(A).$
14. Let $P= \begin{bmatrix}A & B \\ C & D \end{bmatrix}$ be a rectangular matrix with $A$ a square matrix of order $n$ and $\vert A\vert \neq 0.$ Then show that ${\mbox{ rank }}(P) = n$ if and only if $D = C A^{-1} B.$