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GRE Subject Test: Math : Algebra

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Example Questions

Example Question #191 : Algebra

Evaluate the determinant of the following matrix.

$\begin{bmatrix} 5& -3\\ 7& 8 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

Remember, to evaluate the determinant of a matrix use the following:

$\begin{vmatrix} a&b \\ c&d \end{vmatrix}=ad-bc$

The first step would be to write the determinant of the matrix:

$\begin{vmatrix} 5& -3\\ 7&8 \end{vmatrix}$

Now we can evaluate:

$\begin{vmatrix} 5& -3\\ 7&8 \end{vmatrix}=(5)(8)- (-3)(7)=40+21=61$

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Example Question #11 : Matrices

Evaluate the determinant of the following matrix.

$\begin{bmatrix} 2& 4& 3\\ 3& 0& 1\\ 1& 2& 5 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

To find the determinant of a 3 x 3 matrix, we must use the following:

$\begin{vmatrix} a& b& c\\ d& e& f\\ g& h&k \end{vmatrix}=a\begin{vmatrix} e& f\\ h& k \end{vmatrix} -b\begin{vmatrix} d& f\\ g& k \end{vmatrix} + c\begin{vmatrix} d& e\\ g& h \end{vmatrix}$

The first thing we must do is write the determinant:

$\begin{vmatrix} 2& 4& 3\\ 3& 0& 1\\ 1& 2& 5 \end{vmatrix}$

Now we can proceed to evaluate the determinant

$\begin{vmatrix} 2& 4& 3\\ 3& 0& 1\\ 1& 2& 5 \end{vmatrix}=2\begin{vmatrix} 0& 1\\ 2& 5 \end{vmatrix}-4\begin{vmatrix} 3&1 \\ 1& 5\end{vmatrix}+3\begin{vmatrix} 3& 0\\ 1&2 \end{vmatrix}$

Notice that the numbers 2, 4, and 3 are being multiplied by the determinants of the 2x2 matrices so we have:

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Example Question #4 : Determinants

Find the determinant of the matrix: $\begin{vmatrix} -4 & -7 \\ 1 & 2 \end{vmatrix}$

Possible Answers:

Correct answer:

Explanation:

Step 1: We need to recall how to find the determinant of a 2*2 matrix. To find the determinant of a $2\ast2$ matrix, we need to use the equation D=ad-bc , where =Determinant, and a,b,c,d are from the matrix.

Step 2: Identify a,b,c, and d in the original matrix.
a=first number on top row, b=second number on top row (next to a), c=first number on the bottom row, and d is the second number on the bottom row (next to c).

In this matrix, a=, b=, c=, and d=.

Step 3: Substitute the values of a,b,c, and d into the equation to find the determinant of the matrix.

$ad-bc=(-4\cdot 2)-(-7\cdot 1)$ We will simplify the right side.

ad-bc=(-8)-(-7) . We see that there are two negative signs in the middle, which will become a plus sign.

ad-bc=-8+7 . Simplify the right side.

ad-bc=-1 . is the determinant.

The determinant of the matrix is .

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Example Question #5 : Determinants

Find the determinate of Matrix A.

$A=\begin{bmatrix} 8&2 \\ 5& 3 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

Matrix A is given below.

$A=\begin{bmatrix} a&b \\ c& d \end{bmatrix}$

The formula for the determinate of a 2x2 matrix is:

det(A)=ad-bc

Plugging in the values gives us:

det(A)=8*3-5*2=24-10=14

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Example Question #192 : Algebra

Calculate the dot product of the following vectors:

$a=\left \langle c,6, s\right \rangle$

$b=\left \langle c,10,s\right \rangle$

Possible Answers:

c^2+16+s^2

2c+16+2s

2cs+60

60c^2s^2

c^2+60+s^2

Correct answer:

c^2+60+s^2

Explanation:

Write the formula for dot product given $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ and $b=\left \langle b_{1},b_{2},b_{3}\right \rangle$ .

$a\cdot b= a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$

Substitute the values of the vectors to determine the dot product.

$a\cdot b= c(c)+(6)(10)+s(s) = c^2+60+s^2$

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Example Question #1 : Vector Form

Express -i-k in vector form.

Possible Answers:

$\left \langle -1,1,-1\right \rangle$

$\left \langle -1,0,-1\right \rangle$

$\left \langle -1,-1,0\right \rangle$

$\left \langle -1,-1\right \rangle$

$\left \langle 0,-1,-1\right \rangle$

Correct answer:

$\left \langle -1,0,-1\right \rangle$

Explanation:

The correct form of x,y, and z of a vector is represented in the order of i, j, and k, respectively. The coefficients of i,j, and k are used to write the vector form.

$-i-k = -1i+0j-1k = \left \langle -1,0,-1\right \rangle$

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Example Question #1 : Vector Form

Express -10j in vector form.

Possible Answers:

$\left \langle 0,-10,0\right \rangle$

$\left \langle -10,0,0\right \rangle$

$\left \langle 0,0,-10\right \rangle$

$\left \langle 1,-10,1\right \rangle$

$\left \langle -10,-10,-10\right \rangle$

Correct answer:

$\left \langle 0,-10,0\right \rangle$

Explanation:

The x,y, and z of a vector is represented in the order of i, j, and k, respectively. Use the coefficients of i,j, and k to write the vector form.

$-10j=0i-10j+0k = \left \langle 0,-10,0\right \rangle$

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Example Question #1 : Vector Form

Find the vector form of (6,3,1) to (0,1,3) .

Possible Answers:

$\overrightarrow{v}=[6,2,2]$

$\overrightarrow{v}=[-6,-2,-2]$

$\overrightarrow{v}=[6,2,-3]$

$\overrightarrow{v}=[6,2,-2]$

$\overrightarrow{v}=[6,-2,-2]$

Correct answer:

$\overrightarrow{v}=[6,2,-2]$

Explanation:

When we are trying to find the vector form we need to remember the formula which states to take the difference between the ending and starting point.

Thus we would get:

Given (a,b,c) and (d,e,f)

$\overrightarrow{v}=[d-a, e-b, f-c]$

In our case we have ending point at (6,3,1) and our starting point at (0,1,3) .

Therefore we would set up the following and simplify.

$\overrightarrow{v}=[6-0,3-1,1-3]=[6,2,-2]$

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Example Question #25 : Linear Algebra

Find the dot product of the 2 vectors. $\left \langle a,2a\right \rangle \cdot \left \langle a,a\right \rangle$

Possible Answers:

$\left \langle a^2,2a^2\right \rangle$

$\left \langle 3a^2,3a^2\right \rangle$

$\left \langle a,2a\right \rangle$

3a^2

Correct answer:

3a^2

Explanation:

The dot product will give a single value answer, and not a vector as a result.

To find the dot product, use the following formula:

$\left \langle x_1,x_2\right \rangle \cdot \left \langle y_1,y_2\right \rangle=(x_1)(y_1)+(x_2)(y_2)$

$\left \langle a,2a\right \rangle \cdot \left \langle a,a\right \rangle = (a)(a)+(2a)(a)=a^2+2a^2=3a^2$

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Example Question #3 : Vector Form

Assume that Billy fired himself out of a circus cannon at a velocity of $25 \:mph$ at an elevation angle of degrees. Write this in vector component form.

Possible Answers:

$\left \langle23.097, 9.567 \right \rangle$

$\left \langle 25.25,25.25\right \rangle$

$\left \langle 24.78,-3.309\right \rangle$

$\left \langle22.658,10.565 \right \rangle$

$\left \langle 50,50\right \rangle$

Correct answer:

$\left \langle22.658,10.565 \right \rangle$

Explanation:

The firing of the cannon has both x and y components.

Write the formula that distinguishes the x and y direction and substitute.

$\left \langle x,y\right \rangle= \left \langle vcos(\theta),vsin(\theta)\right \rangle$

Ensure that the calculator is in degree mode before you solve.

$\left \langle 25cos25,25sin25\right \rangle = \left \langle22.658,10.565 \right \rangle$

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GRE Subject Test: Math : Algebra

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

All GRE Subject Test: Math Resources

Example Questions

Example Question #191 : Algebra

Example Question #11 : Matrices

Example Question #4 : Determinants

Example Question #5 : Determinants

Example Question #192 : Algebra

Example Question #1 : Vector Form

Example Question #1 : Vector Form

Example Question #1 : Vector Form

Example Question #25 : Linear Algebra

Example Question #3 : Vector Form

All GRE Subject Test: Math Resources

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