Mathematics

Inverses in Mathematics

106 Questions

Inverses in mathematics cover additive, multiplicative, and matrix operations. These concepts test the ability to reverse mathematical functions and transformations. Such questions frequently appear in various competitive examinations to evaluate fundamental algebraic skills.

Additive inverseMultiplicative inverseMatrix inversionInverse functionsLinear transformations

Inverses in Mathematics Questions

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

$p=$ $\begin{bmatrix}
0 & x &0 \
 0& 0 & 1
\end{bmatrix}$, then $p^{-1}$=


  1. Not possible to get an inverse
  2. $\begin{bmatrix}

    x & -a &-bx \\

    0&1 &0 \\

    0&0 &x

    \end{bmatrix}$
  3. $\mathrm{x}$ $\begin{bmatrix}

    x & -a &-bx \\

    0&1 &0 \\

    0&0 &x

    \end{bmatrix}$
  4. $x^{2} \begin{bmatrix}

    x & -a &-bx \\

    0&1 &0 \\

    0&0 &x

    \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Requirements to have an Inverse

1. The matrix must be square (same number of rows and columns).
2. The determinant of the matrix must not be zero.
It doesn't satisfy the first condition  so inverse of the given matrice $\left[ \begin{matrix} 0 & x & 0 \\ 0 & 0 & 1 \end{matrix} \right] $ is not possible as it is not a square matrice

option A is correct.
Multiple choice maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

Consider two matrix $A = \begin{bmatrix} 1 & 2\ 2 & 1\ 1 & 1 \end{bmatrix}$ and $ B = \begin{bmatrix} 1 & 2 & -4\  2 & 1 & -4 \end{bmatrix}$. Which one of the following is correct ?

  1. B is the right inverse of A

  2. B is the left inverse of A

  3. B is the both sided inverse of A

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For two $m \times  n$ and $n \times m$ matrices $A$ and $B$

If $AB =I$, then $B$ is right inverse of $A$
If $BA=I$, then $B$ is left inverse of $A$
Note that if $AB=BA=I$, $B$ is both sided inverse of $A$
If $m=n$, which isn't true here
$AB=\begin{bmatrix} 1 & 2 \ 2 & 1 \ 1 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 & -4 \ 2 & 1 & -4 \end{bmatrix}=\begin{bmatrix} 5 & 4 & -4 \ 4 & 5 & -4 \ 3 & 3 & 0 \end{bmatrix}\neq I$
$BA=\begin{bmatrix} 1 & 2 & -4 \ 2 & 1 & -4 \end{bmatrix}\begin{bmatrix} 1 & 2 \ 2 & 1 \ 1 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}=I$
$\therefore$ B is left inverse of A

Multiple choice

The additive inverse of a negative integer

  1. is always negative

  2. is always positive

  3. is the same integer

  4. Zero
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The additive inverse of a number x is -x. If x is negative, -x is positive (e.g., the inverse of -5 is 5).

Multiple choice

Additive inverse of -a is

  1. a

  2. 1/a

  3. -a

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The additive inverse of a number x is -x. Therefore, the additive inverse of -a is -(-a), which equals a.

Multiple choice maths operations adding and subtracting numbers using place value addition & subtraction mental additions and subtractions

The additive inverse of $\sqrt { 3 } -2\quad is$_____________

  1. $\sqrt { 3 } +2$
  2. $-\sqrt { 3 } +2$
  3. $-\sqrt { 3 } -2$
  4. $\sqrt { 3 } +\sqrt { 2 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
The additive inverse of a number a is the number that, when added to a, yields zero. 
This number is also known as the opposite (number), sign change, and negation.

$Additive \ Inverse \ of (\sqrt {3}-2 )\ is$
$\rightarrow -(\sqrt{3}-2)$
$\rightarrow -\sqrt{3}+2$
$\rightarrow B \ is \ correct \ answer$
Multiple choice

What is the inverse of the function (f(x) = 2x + 1)?

  1. \(f^{-1}(x) = \frac{x - 1}{2}\)
  2. \(f^{-1}(x) = \frac{x + 1}{2}\)
  3. \(f^{-1}(x) = 2x - 1\)
  4. \(f^{-1}(x) = -2x + 1\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The inverse of a function (f(x)) is the function (f^{-1}(x)) such that (f(f^{-1}(x)) = x) and (f^{-1}(f(x)) = x). To find the inverse of (f(x) = 2x + 1), we can solve for (x) in terms of (f(x)): (f(x) = 2x + 1) (\Rightarrow 2x = f(x) - 1) (\Rightarrow x = \frac{f(x) - 1}{2}) (\therefore f^{-1}(x) = \frac{x - 1}{2}).

Multiple choice

The inverse Laplace transform of $F(s) = \frac{1}{s^2 + 4}$ is:

  1. $f(t) = \frac{1}{2} \sin(2t)$
  2. $f(t) = \frac{1}{2} \cos(2t)$
  3. $f(t) = \frac{1}{2} e^{2t}$
  4. $f(t) = \frac{1}{2} e^{-2t}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The inverse Laplace transform of $\frac{1}{s^2 + 4}$ is $\frac{1}{2} \sin(2t)$.

Multiple choice

What is the matrix representation of the linear transformation f(x) = 2x - 1?

  1. [[2, 0], [0, -1]]

  2. [[2, 1], [0, -1]]

  3. [[2, -1], [0, 1]]

  4. [[2, 0], [1, -1]]
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

The matrix representation of a linear transformation is a matrix whose columns are the coordinates of the transformed basis vectors. In this case, the basis vector is [1, 0], and its transformed vector is [2, -1]. Therefore, the matrix representation is [[2, -1]].

Multiple choice

What is the matrix representation of the linear transformation f(x) = 2x - 1?

  1. [[2, 0], [0, -1]]

  2. [[2, 1], [0, -1]]

  3. [[2, -1], [0, 1]]

  4. [[2, 0], [1, -1]]
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The matrix representation of a linear transformation is the matrix that performs the same transformation when multiplied by a column vector. In this case, the matrix representation of f(x) = 2x - 1 is [[2, -1], [0, 1]] because [2, -1][x] = 2x - 1 and [0, 1][y] = y for any scalars x and y.

Multiple choice

Which of the following is an example of a linear transformation that is not invertible?

  1. f(x) = 2x + 1

  2. f(x) = x^2

  3. f(x) = sin(x)

  4. f(x) = |x|
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A linear transformation is invertible if there exists a linear transformation g(x) such that f(g(x)) = g(f(x)) = x for all vectors x in the domain. In this case, f(x) = x^2 is not invertible because there is no linear transformation g(x) such that f(g(x)) = g(f(x)) = x for all vectors x in the domain.

Multiple choice

What is the nullity of the linear transformation f(x) = [x1, x2, x3]?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The nullity of a linear transformation is the dimension of the kernel of the transformation. In this case, the kernel of f(x) = [x1, x2, x3] is the set of all vectors [x1, x2, x3] such that x1 = x2 = x3 = 0, which is a zero-dimensional subspace of R^3. Therefore, the nullity of f(x) is 0.

Multiple choice

What is the inverse of the function (f(x) = 3x - 2)?

  1. \(f^{-1}(x) = \frac{x + 2}{3}\)
  2. \(f^{-1}(x) = \frac{x - 2}{3}\)
  3. \(f^{-1}(x) = \frac{1}{3}x + 2\)
  4. \(f^{-1}(x) = \frac{1}{3}x - 2\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The inverse of a function is the function that undoes the original function.

Multiple choice

What is the inverse Laplace transform of $\frac{1}{s^2+a^2}$?

  1. $\frac{1}{a}\sin(at)$
  2. $\frac{1}{a}\cos(at)$
  3. $\frac{1}{2a}\sin(at)$
  4. $\frac{1}{2a}\cos(at)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The inverse Laplace transform of $\frac{1}{s^2+a^2}$ is $\frac{1}{a}\sin(at)$.