List of top Mathematics Questions on Algebra

The equation $\left|\begin{matrix}2& 1& 1\\ 1& 1&-1\\ y& x^{2}& x\end{matrix}\right|=0$ represents a parabola passing through the points}

If $W=\{\left(\begin{matrix}x& y\\ z& 0\end{matrix}\right):x,y,z\in R\}$ is a subspace of the vector space $M_{2}$ of $2\times2$ matrices over the field of real numbers R, then $dimW=_$

If T is a linear transformation from $R^{2}\rightarrow R^{2}$ defined by $T(1,-1)=(2,-4)$ and $T(1,1)=(0,2)$. Then $T(a,b)=$

Let $T:R^{3}\rightarrow R^{3}$ be a linear transformation defined by $T(a,b,c)=(a+b-c, a+b+c, b-c)$. Then the matrix of $T$ with respect to the ordered basis $B = \{(0,1,0), (0,0,1), (1,0,0)\}$ is}

If $T:R^{3}\rightarrow R^{2}$ be a linear transformation defined by $T(a,b,c)=(a+b+c,a-b-c)$, then which of the following is an element in the null space of T.

Which of the following sets of vectors in $R^{2}$ are linearly independent over R.

The coordinates of (4,5,6) with respect to the basis set $e_{1}=(1,1,1), e_{2}=(-1,1,0), e_{3}=(1,0,-1)$ are}

Let $A=\left[\begin{matrix}1& 1& 1\\ 2& 2& 3\\ x& y& 2\end{matrix}\right]$ and let $V=\{(x,y,z)\in R^{3}:\det(A)=0\}$. Then the dimension of V equals}

Which of the following polynomial is irreducible over $Z_{3}$, a field modulo 3?

Let $f(x)=2+3x^{2}, g(x)=1+2x^{2}$ be polynomials in the ring $Z_{4}$ under mod 4. Then, $deg\{f(x)g(x)\}=$

The ring of all $2 \times 2$ matrices over reals is

Let Z be the ring of integers and $f: Z \to 2Z$ defined by $f(n)=2n$, $\forall n \in Z$. Then f is

Invertible elements in the ring of integers are

The ring of integers (Z, +, $\cdot$) is

Choose the WRONG statement

Consider the group homomorphism $f: (Z, +) \to (G = \{i^n : n \in Z\}, \cdot)$ defined by $f(n) = i^n, \forall n \in Z$. Then, $Ker f = $

Let H be a subgroup of a group G, where $O(H) = m$ and $O(G) = n$. Then, for any $a \in G$, $O(aH) = $

The permutations $f = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 8 & 2 & 5 & 1 & 3 & 7 & 6 \end{pmatrix}, g = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 5 & 1 & 2 & 8 & 7 & 4 & 3 & 6 \end{pmatrix}$ of $S_8$ are respectively}

Consider the following statements:
A. Every finite cyclic group of order n is isomorphic to the group $(Z_n, +_n)$.
B. Every infinite cyclic group is isomorphic to the group (Z, +).
Which of the following statement is TRUE?

The intersection of the subgroups 2Z, 3Z and 5Z, i.e., $2Z \cap 3Z \cap 5Z$ of (Z, +) is

A noncyclic group among the following is

On the set of integers Z, the operation defined as following is not a binary operation.

In the group $(R-\{-1\}, *)$ where $*$ is defined by $a*b = a+b+ab, \forall a,b \in R-\{-1\}$, inverse of the element 2 is}

Express the matrix $A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$ as a sum of symmetric and skew-symmetric matrices.

If \(\alpha>0\) and \(\beta\) are the roots of the equation \[ \sqrt{\frac{x-1}{x+2}} + \sqrt{\frac{x+2}{x-1}} = \frac{13}{6}, \] then \(4\alpha-\beta=\)