List of practice Questions

Evaluate the determinants in Exercises 1 and 2. 
(i) \(\begin{vmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{vmatrix}\) 

(ii) \(\begin{vmatrix}x^2&-x+1&x-1\\& x+1&x+1\end{vmatrix}\)

Evaluate the determinants in Exercises 1 and 2. 

 \(\begin{vmatrix}2&4\\-5&-1\end{vmatrix}\)

Let f :X\(\to\) Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g₁ and g₂ are two inverses of f. Then for all y∈ Y, fog₁(y) = I_Y (y) = fog₂ (y). Use one-one ness of f).

Show that f :[−1,1]→R,given by f(x)= \(\frac {x}{x+2}\) is one-one. Find the inverse of the function f :[−1,1] \(\to\) Range f.
(Hint: For y ∈Range f, y = f(x)= \(\frac {x}{x+2}\) , for some x in [−1, 1], i.e.,x= \(\frac {2y}{1-y}\)

Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix}2&1\\1&1\end{bmatrix}\)

Find the transpose of each of the following matrices:
  I.\(\begin{bmatrix}5\\\frac{1}{2}\\-1\end {bmatrix}\)

  II.\(\begin{bmatrix}1&-1\\2&3\end{bmatrix}\)

  III.\(\begin{bmatrix}-1&5&6\\\sqrt3&5&6\\2&3&-1\end{bmatrix}\)

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Assume X,Y,Z,W and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3, and p x k respectively.
The restriction on n, k and p so that PY+WY will be defined are: 
 A. k = 3,p = n 
 B. k is arbitrary, p = 2 
 C. p is arbitrary, k=3 
 D. k=2,p=3

Assume X,Y,Z,W and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3, and p x k respectively. If n = p, then the order of the matrix 7X-5Zis 

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

Which of the given values of x and y make the following pair of matrices equal \(\begin{bmatrix}3x+y&5\\y+1&2-3x\end{bmatrix}=\begin{bmatrix}0&y-2\\8&4\end{bmatrix}\)

A=\([a_{ji}]_{m*n}\) is a square matrix, if

Find the value of a,b,c, and d from the equation: \(\begin{bmatrix}a-b&2a+c\\2a-b&3c+d\end{bmatrix}=\begin{bmatrix}-1&5\\0&13\end{bmatrix}\)

Find the value of x, y, and z from the following equation:
 I.\(\begin{bmatrix} 4&3&\\x&5\end{bmatrix}=\begin{bmatrix}y&z\\1&5\end{bmatrix}\)

  II. \(\begin{bmatrix}x+y&2\\5+z&xy\end{bmatrix}=\begin{bmatrix}6&2\\5&8\end{bmatrix}\)

   III. \(\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}\)

Check the injectivity and surjectivity of the following functions:

1. f: N \(\to\) N given by f(x) = x²
2. f: Z \(\to\) Z given by f(x) = x²
3. f: R \(\to\) R given by f(x) = x²
4. f: N \(\to\) N given by f(x) = x³
5. f: Z \(\to\) Z given by f(x) = x³

Let f : R \(\to\) R be the Signum Function defined as \(f(x) =   \begin{cases}     1,     & \quad \text x>0 \\     0,  & \quad x=0 \\  -1, &\quad x<0 \end{cases}\)

and \(g: R \to R\) be the Greatest Integer Function given by g (x)= [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0,1]?

Define a binary operation *on the set {0,1,2,3,4,5} as
\(a*b =   \begin{cases}     a+b       & \quad \text{if } a+b<6 \\     a+b-6  & \quad \text{if } a+b\geq6    \end{cases}\)
Show that zero is the identity for this operation and each element a≠0 of the set is invertible with 6−a being the inverse of a.

Consider the binary operations*: R ×R → and o: R×R →R defined as \(a*b=\mid a-b \mid \)and a o b = a,∀a,b∈R. 
Show that * is commutative but not associative, o is associative but not commutative. 
Further, show that ∀a,b,c∈R, a * (b o c)= (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.