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

Given an orthogonal matrix $\begin{bmatrix} \ 1 & 1 & 1 & 1 \ \ 1 & 1 & -1 & -1 \ \ 1 & -1 & 0 & 0 \ \ 0 & 0 & 1 & -1 \end{bmatrix}$$\begin{bmatrix} \ AA^T \ \end{bmatrix}^{-1}$ is
  1. $\begin{bmatrix} \ \frac{1}{4} & 0 & 0 & 0 \\ \ 0 & \frac{1}{4} & 0 & 0 \\ \ 0 & 0 & \frac{1}{2} & 0 \\ \ 0 & 0 & 0 & \frac{1}{2} \\ \end{bmatrix}$
  2. $\begin{bmatrix} \ \frac{1}{2} & 0 & 0 & 0 \\ \ 0 & \frac{1}{2} & 0 & 0 \\ \ 0 & 0 & \frac{1}{2} & 0 \\ \ 0 & 0 & 0 & \frac{1}{2} \\ \end{bmatrix}$
  3. $\begin{bmatrix} \ 1 & 0 & 0 & 0 \\ \ 0 & 1 & 0 & 0 \\ \ 0 & 0 & 1 & 0 \\ \ 0 & 0 & 0 & 1 \\ \end{bmatrix}$
  4. $\begin{bmatrix} \ \frac{1}{4} & 0 & 0 & 0 \\ \ 0 & \frac{1}{4} & 0 & 0 \\ \ 0 & 0 & \frac{1}{4} & 0 \\ \ 0 & 0 & 0 & \frac{1}{4} \\ \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\text{From orthogonal matrix} \\ \begin{bmatrix} \ AA^T \ \end{bmatrix} = I \\ \text{Since the inverse of I is I, thus} \\ \begin{bmatrix} \ AA^T \ \end{bmatrix} = I^{-1} = I $

Multiple choice

What is the additive inverse of (3/7) + (- 6/11) + (- 16/42) + (10/44)?
    • 125/462
    • 12/22

  3. 30/132

  4. 125/462

  5. 250/462
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

(3/7) + (- 6/11) + (- 16/42) + (10/44) For finding the additive inverse of the above expression, we need to find the value of the above expression. Additive inverse of a number is negative value of the same number. So, addition of these two numbers will be zero. For example, additive inverse of 1 is - 1. Now, the value of the given expression: 3/7 - 6/11 - 16/42 + 10/42 = 3/7 - 6/11 - 8/21 + 5/22 = 198/462 - 252/462 - 176/462 + 105/462 = (198 - 252 - 176 + 105) / 462 = - 125/462

Therefore, additive inverse of the given expression is (125/462) so that 125/ 462 - 125/462 = 0.

Multiple choice

What is the additive inverse of the following expression?

(1/2 + 1/3 + ¼ ) / (½ - 1/3)
    • 13/2

  2. 13/2

  3. 1/6

  4. 13/12
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

(1/2 + 1/3 + ¼ ) / (½ - 1/3) For finding the additive inverse of the above expression, we need to find the value of the above expression.

Additive inverse of a number is negative value of the same number. So, addition of these two numbers will be zero. For example, additive inverse of 1 is - 1 (As 1 - 1 = 0).

Now, the value of the given expression is (1/2 + 1/3 + ¼ ) / ( ½ - 1/3)

= {(6 + 4 + 3) / 12} / {( 3-2)/6}

= {13/12} / {1/6}

= 13/2

Therefore, additive inverse of the given expression is - 13/2 so that 13/2 - 13/2 = 0.

Multiple choice

What is multiplicative inverse of the following expression?

{ ½ - 2/4 + 1/3 - ¼ } / {½ - 2/4 + 1/3 + ¼ } ?
    • 1/7

  2. 7

  3. 1/7

  4. 49
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Multiplicative inverse of a number is equal to the reciprocal of the given number such that product of the given number and its reciprocal is equal to 1. The given expression is { ½ - 2/4 + 1/3 - ¼ } / { ½ - 2/4 + 1/3 + ¼ } = { ½ - ½ + 1/3 - ¼ } / { ½ - ½ + 1/3 + ¼ } = { 1/3 - ¼ } / {1/3 + ¼ } = {1/12} / {7/12} = 1/7

Therefore, multiplicative inverse of the given number is 7.

Multiple choice

Directions: Carry Two Marks Each.

For the matrix P = $\begin{bmatrix} \ 4+3i & -i \ \ i & 4-3i \ \end{bmatrix}$, where i = $\sqrt{-1}$, the inverse of matrix P is
  1. $\frac{1}{24}\begin{bmatrix} \ 4-3i & i \\ \ -i & 4+3i \\ \end{bmatrix}$
  2. $\frac{1}{25}\begin{bmatrix} \ i & 4-3i \\ \ 4+ 3i & -i \\ \end{bmatrix}$
  3. $\frac{1}{24}\begin{bmatrix} \ 4+3i & -i \\ \ i & 4-3i \\ \end{bmatrix}$
  4. $\frac{1}{25}\begin{bmatrix} \ 4+3i & -i \\ \ i & 4-3i \\ \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\text{For a given matrix } P=\begin{bmatrix} \ 4+3i & -i \ \ i & 4-3i \ \end{bmatrix}\\ where,\\ \hspace{1.5cm} i=\sqrt{-1,}\text{the inverse of matrix P=?} $

Multiple choice

The inverse of the 2 x 2 matrix $\begin{bmatrix} \ 1 & 2 \ \ 5 & 7 \ \end{bmatrix}$ is
  1. $\frac{1}{3}\begin{bmatrix} \ -7 & 2 \\ \ 5 & -1 \\ \end{bmatrix}$
  2. $\frac{1}{3}\begin{bmatrix} \ 7 & 2 \\ \ 5 & 1 \\ \end{bmatrix}$
  3. $\frac{1}{3}\begin{bmatrix} \ 7 & -2 \\ \ -5 & 1 \\ \end{bmatrix}$
  4. $\frac{1}{3}\begin{bmatrix} \ -7 & -2 \\ \ -5 & -1 \\ \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The inverse of a 2x2 matrix [[a, b], [c, d]] is (1/det) * [[d, -b], [-c, a]]. The determinant is (1*7 - 2*5) = 7 - 10 = -3. The inverse is (1/-3) * [[7, -2], [-5, 1]], which simplifies to (1/3) * [[-7, 2], [5, -1]].

Multiple choice reciprocal equations theory of equations maths

The inverse of the function $f(x) = \frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}}$ is

  1. $\dfrac{1}{2}\ell n\dfrac{{1 + x}}{{1 - x}}$
  2. $\dfrac{1}{2}\ell n\dfrac{{2 + x}}{{2 - x}}$
  3. $\dfrac{1}{2}\ell n\dfrac{{1 - x}}{{1 + x}}$
  4. $2\ell n(1 + x)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$y=\cfrac { { e }^{ x }-{ e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } }$

$y=\cfrac { \cfrac { { e }^{ 2x }-1 }{ { e }^{ x } }  }{ \cfrac { { e }^{ 2x }+1 }{ { e }^{ x } }  }$
 $y=\cfrac { { e }^{ 2x }-1 }{ { e }^{ 2x }+1 }$
 for universe replace;$x$ with $y$
$\cfrac { x }{ 1 } =\cfrac { { e }^{ 2y }-1 }{ { e }^{ 2y }+1 }$
 using componendo divodendo
 $\cfrac { x+1 }{ x-1 } =\cfrac { { e }^{ 2y }-1+{ e }^{ 2y }+1 }{ { e }^{ 2y }-1-{ e }^{ 2y }-1\quad  }$
 $\cfrac { x+1 }{ x-1 } =-\cfrac { 2{ e }^{ 2y } }{ 2 }$
 $\cfrac { 1+x }{ 1-x } ={ e }^{ 2y }$
$\left(\cfrac { 1+x }{ 1-x } \right)=2y$
$ y=\cfrac { 1 }{ 2 } ln\left(\cfrac { 1+x }{ 1-x } \right)$

Multiple choice maths bar graphs histograms for non-uniform class widths histograms with unequal class intervals revisiting histograms

Strictly monotonical function guarantee inverse mapping as

  1. single valued

  2. multi valued

  3. dual valued

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

A strictly monotonic function is always injective (one-to-one). This ensures that for every output in the range, there is exactly one corresponding input in the domain, making the inverse mapping single-valued.

Multiple choice business maths inverse of a matrix and linear equations non singular matrix singular & non-singular matrix determinants

If the matrix $\begin{bmatrix} \alpha  & 2 & 2 \ -3 & 0 & 4 \ 1 & -1 & 1 \end{bmatrix}$ is not invertible, then:

  1. $\alpha =-5$
  2. $\alpha =5$
  3. $\alpha =-0$
  4. $\alpha =-1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a matrix to be not invertible, determinant of the matrix should be $0$
So, $\left| \begin{matrix} \alpha  & 2 & 2 \ -3 & 0 & 4 \ 1 & -1 & 1 \end{matrix} \right| =0$

$\Rightarrow -(-3)(2-(-2))-4(-\alpha -2)=0\ \Rightarrow 12+4\alpha +8=0\ \Rightarrow 4\alpha =-20\ \Rightarrow \alpha =-5$
Hence, option A is correct