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GRE Subject Test: Math : Inverses

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Example Question #13 : Linear Algebra

Which of the following is the inverse of h(x) ?

h(x)=3lnx

Possible Answers:

$y=3e^{x}$

$y=e^{\frac{x}{3}}$

$y=\frac{e^x}{3}$

$y=9e^{x}$

Correct answer:

$y=e^{\frac{x}{3}}$

Explanation:

Which of the following is the inverse of h(x) ?

h(x)=3lnx

To find the inverse of a function, we need to swap x and y, and then rearrange to solve for y. The inverse of a function is basically the function we get if we swap the x and y coordinates for every point on the original function.

So, to begin, we can replace the h(x) with y.

y=3lnx

Next, swap x and y

x=3lny

Now, we need to get y all by itself; we can to begin by dividng the three over.

$\frac{1}{3}x=lny$

Now, recall that

And that we can rewrite any log as an exponent as follows:

log_b(a)=x

b^x=a

So with that in mind, we can rearrange our function to get y by itself:

$\frac{1}{3}x=lny$

Becomes our final answer:

$y=e^{\frac{x}{3}}$

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Example Question #14 : Linear Algebra

Find the Inverse of Matrix B $(B^{-1})$ where

$B=\begin{bmatrix} 2&4 \\ 5&8 \end{bmatrix}$ .

Possible Answers:

$B^{-1}=\frac{-1}{4}\begin{bmatrix} 8&-4 \\ -5& 2 \end{bmatrix}$

$B^{-1}=\frac{-1}{4}\begin{bmatrix} 2&-4 \\ -5& 8 \end{bmatrix}$

$B^{-1}=\frac{1}{4}\begin{bmatrix} 2&4 \\ 5& 8 \end{bmatrix}$

$B^{-1}=\frac{-1}{4}\begin{bmatrix} 8&4 \\ 5& 2 \end{bmatrix}$

$B^{-1}=\frac{1}{4}\begin{bmatrix} 8&-4 \\ -5& 2 \end{bmatrix}$

Correct answer:

$B^{-1}=\frac{-1}{4}\begin{bmatrix} 8&-4 \\ -5& 2 \end{bmatrix}$

Explanation:

To find the inverse matrix of B use the following formula,

$B^{-1}=\frac{1}{ad-bc}\begin{bmatrix} d&-b \\ -c& a \end{bmatrix}$ .

Since the matrix B is given as,

$B=\begin{bmatrix} 2&4 \\ 5&8 \end{bmatrix}$

the inverse becomes,

$B^{-1}=\frac{1}{2*8-5*4}\begin{bmatrix} 8&-4 \\ -5& 2 \end{bmatrix}$

$B^{-1}=\frac{-1}{4}\begin{bmatrix} 8&-4 \\ -5& 2 \end{bmatrix}$ .

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Example Question #15 : Linear Algebra

Find the inverse of the following matrix, if possible.

$A=\begin{bmatrix} 4&-4 \\ -4&-4 \end{bmatrix}$

Possible Answers:

The inverse does not exist.

$\begin{bmatrix} \frac{1}{8}&-\frac{1}{8} \\ -\frac{1}{8}& -\frac{1}{8} \end{bmatrix}$

$-\frac{1}{32}$

$\begin{bmatrix} \frac{1}{4}&-\frac{1}{4} \\ -\frac{1}{4}& -\frac{1}{4} \end{bmatrix}$

Correct answer:

$\begin{bmatrix} \frac{1}{8}&-\frac{1}{8} \\ -\frac{1}{8}& -\frac{1}{8} \end{bmatrix}$

Explanation:

Write the formula to find the inverse of a matrix.

$\begin{bmatrix} a&b\\ c&d \end{bmatrix}^-^1=\frac{1}{ad-bc}\begin{bmatrix} d& -b\\ -c& a \end{bmatrix}$

Substituting in the given matrix we are able to find the inverse matrix.

$\begin{bmatrix} 4&-4 \\ -4&-4 \end{bmatrix}^-^1=\frac{1}{(4)(-4)-(-4)(-4)}\begin{bmatrix} -4&4 \\ 4&4 \end{bmatrix}$

$\frac{1}{-32}\begin{bmatrix} -4&4 \\ 4&4 \end{bmatrix}=\begin{bmatrix} \frac{1}{8}&-\frac{1}{8} \\ -\frac{1}{8}& -\frac{1}{8} \end{bmatrix}$

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GRE Subject Test: Math : Inverses

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #13 : Linear Algebra

Example Question #14 : Linear Algebra

Example Question #15 : Linear Algebra

All GRE Subject Test: Math Resources

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