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SAT II Math II : SAT Subject Test in Math II

Study concepts, example questions & explanations for SAT II Math II

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All SAT II Math II Resources

2 Diagnostic Tests 130 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : Matrices

Simplify:

$\begin{bmatrix} 14&22\\ -3 &4 \end{bmatrix}+\begin{bmatrix} -3&12\\ 5 &-10 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 13\\ 28 \end{bmatrix}$

$\begin{bmatrix} 11&34\\ 2 &-6 \end{bmatrix}$

$\begin{bmatrix} 17 & -5 \\ 3 & 1 \end{bmatrix}$

$\begin{bmatrix} 17&10\\ -8 &14 \end{bmatrix}$

$\begin{bmatrix} 45&-4 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 11&34\\ 2 &-6 \end{bmatrix}$

Explanation:

Matrix addition is very easy! All that you need to do is add each correlative member to each other. Think of it like this:

$\begin{bmatrix} 14&22\\ -3 &4 \end{bmatrix}+\begin{bmatrix} -3&12\\ 5 &-10 \end{bmatrix}=\begin{bmatrix} 14-3&22+12\\ -3+5 &4-10 \end{bmatrix}$

Now, just simplify:

$\begin{bmatrix} 11&34\\ 2 &-6 \end{bmatrix}$

There is your answer!

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Example Question #13 : Find The Sum Or Difference Of Two Matrices

Simplify:

$\begin{bmatrix} -5&7\\ 102 & -12 \end{bmatrix}+\begin{bmatrix} 3&107\\ -10 & -9 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} -2&114\\ 92 & -21 \end{bmatrix}$

$\begin{bmatrix} 90\\ 93 \end{bmatrix}$

$\begin{bmatrix} -8&-100\\ 112 & -3 \end{bmatrix}$

$\begin{bmatrix} 112\\ 71 \end{bmatrix}$

$\begin{bmatrix} 112 & 71\end{bmatrix}$

Correct answer:

$\begin{bmatrix} -2&114\\ 92 & -21 \end{bmatrix}$

Explanation:

Matrix addition is really easy—don't overthink it! All you need to do is combine the two matrices in a one-to-one manner for each index:

$\begin{bmatrix} -5&7\\ 102 & -12 \end{bmatrix}+\begin{bmatrix} 3&107\\ -10 & -9 \end{bmatrix}=\begin{bmatrix} -5+3&7+107\\ 102-10 & -12-9 \end{bmatrix}$

Then, just simplify all of those simple additions and subtractions:

$\begin{bmatrix} -5+3&7+107\\ 102-10 & -12-9 \end{bmatrix}=\begin{bmatrix} -2&114\\ 92 & -21 \end{bmatrix}$

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Example Question #1 : Matrices

Evaluate: $-3\begin{bmatrix} 2& 3 & 4\\ 4&-5 & 10 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} -12& -1\\ 15& 0\\ -30& 1 \end{bmatrix}$

$\begin{bmatrix} -12&-6 \\ 15& -9\\ -30&-12 \end{bmatrix}$

$\begin{bmatrix} -1& 0& 1\\ -12& 15&-30 \end{bmatrix}$

$\begin{bmatrix} -18& 6& -42 \end{bmatrix}$

$\begin{bmatrix} -6& -9& -12\\ -12& 15& -30 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} -6& -9& -12\\ -12& 15& -30 \end{bmatrix}$

Explanation:

This problem involves a scalar multiplication with a matrix. Simply distribute the negative three and multiply this value with every number in the 2 by 3 matrix. The rows and columns will not change.

$-3\begin{bmatrix} 2& 3 & 4\\ 4&-5 & 10 \end{bmatrix}=\begin{bmatrix} -6&-9 &-12 \\ -12& 15&-30 \end{bmatrix}$

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Example Question #1 : How To Find Scalar Interactions With A Matrix

Simplify:

$\begin{bmatrix} 13 & 4\\ 71 &3 \end{bmatrix} + 4\begin{bmatrix} 5 & 3\\ 11 &22 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 31 & 93\\ 3 &21 \end{bmatrix}$

$\begin{bmatrix} 49\\ 206\end{bmatrix}$

$\begin{bmatrix} 33 & 16\\ 115 & 91\end{bmatrix}$

$\begin{bmatrix} 49 & 206\end{bmatrix}$

$\begin{bmatrix} 20 & 12\\ 44 &88 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 33 & 16\\ 115 & 91\end{bmatrix}$

Explanation:

Scalar multiplication and addition of matrices are both very easy. Just like regular scalar values, you do multiplication first:

$\begin{bmatrix} 13 & 4\\ 71 &3 \end{bmatrix} + 4\begin{bmatrix} 5 & 3\\ 11 &22 \end{bmatrix}= \begin{bmatrix} 13 & 4\\ 71 &3 \end{bmatrix}+\begin{bmatrix} 20 & 12\\ 44 &88 \end{bmatrix}$

The addition of matrices is very easy. You merely need to add them directly together, correlating the spaces directly.

$\begin{bmatrix} 13 & 4\\ 71 &3 \end{bmatrix} + \begin{bmatrix} 20 & 12\\ 44 &88 \end{bmatrix} = \begin{bmatrix} 13 +20 & 4+12\\ 71+44 &3 +88\end{bmatrix}$

$\begin{bmatrix} 13 +20 & 4+12\\ 71+44 &3 +88\end{bmatrix}= \begin{bmatrix} 33 & 16\\ 115 & 91\end{bmatrix}$

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Example Question #1061 : Algebra

$5\begin{bmatrix} a \\ b \end{bmatrix} = 14\begin{bmatrix} 20 \\ 12 \end{bmatrix}$

What is a+b ?

Possible Answers:

$\frac{191}{4}$

$\frac{448}{5}$

$\frac{148}{5}$

$\frac{41}{5}$

Correct answer:

$\frac{448}{5}$

Explanation:

You can begin by treating this equation just like it was:

5x=14y

That is, you can divide both sides by :

$\begin{bmatrix} a \\ b \end{bmatrix} = \frac{14}{5}\begin{bmatrix} 20 \\ 12 \end{bmatrix}$

Now, for scalar multiplication of matrices, you merely need to multiply the scalar by each component:

$\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} \frac{14}{5}*20 \\ \frac{14}{5}*12 \end{bmatrix}$

Then, simplify:

$\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 56 \\ \frac{168}{5} \end{bmatrix}$

Therefore, $a+b=56+\frac{168}{5}=\frac{280}{5}+\frac{168}{5}=\frac{448}{5}$

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Example Question #4 : Find The Sum Or Difference Of Two Matrices

Given the following matrices, what is the product of and ?

$\begin{bmatrix} 1&-4 \\ 2&x \end{bmatrix} - \begin{bmatrix} y&4 \\ 8&1 \end{bmatrix} = \begin{bmatrix} -8&-8 \\ -6&4 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

When subtracting matrices, you want to subtract each corresponding cell.

$\begin{bmatrix} 1-y&-4-4 \\ 2-8&x-1 \end{bmatrix}= \begin{bmatrix} -8&-8 \\ -6&4 \end{bmatrix}$

Now solve for and

x-1=4

x=5

1-y=-8

y=9

xy = 45

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Example Question #1 : How To Subtract Matrices

If $\begin{bmatrix} a&b\\ c&d \end{bmatrix} + \begin{bmatrix} 3&14\\ 12&4 \end{bmatrix} = \begin{bmatrix} 19&2\\ 4&5 \end{bmatrix}$ , what is a+b+c+d ?

Possible Answers:

Correct answer:

Explanation:

You can treat matrices just like you treat other members of an equation. Therefore, you can subtract the matrix

$\begin{bmatrix} 3&14\\ 12&4 \end{bmatrix}$

from both sides of the equation. This gives you:

$\begin{bmatrix} a&b\\ c&d \end{bmatrix}=\begin{bmatrix} 19&2\\ 4&5 \end{bmatrix}-\begin{bmatrix} 3&14\\ 12&4 \end{bmatrix}$

Now, matrix subtraction is simple. You merely subtract each element, matching the correlative spaces with each other:

$\begin{bmatrix} a&b\\ c&d \end{bmatrix} = \begin{bmatrix} 19-3&2-14\\ 4-12&5-4 \end{bmatrix}$

Then, you simplify:

$\begin{bmatrix} 19-3&2-14\\ 4-12&5-4 \end{bmatrix} = \begin{bmatrix} 16&-12\\ -8&1 \end{bmatrix}$

Therefore, a+b+c+d = 16-12-8+1=-3

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Example Question #1062 : Algebra

If $\frac{1}{2}$\bigl(\begin{smallmatrix} 2a\\4b \end{smallmatrix} \bigr)$=$\bigl(\begin{smallmatrix} 53 \\ 98 \end{smallmatrix} \bigr)$$ , what is a+b ?

Possible Answers:

$\frac{103}{2}$

$\frac{99}{2}$

102

Correct answer:

102

Explanation:

Begin by distributing the fraction through the matrix on the left side of the equation. This will simplify the contents, given that they are factors of :

$\frac{1}{2}\begin{bmatrix} 2a\\4b \end{bmatrix}=\begin{bmatrix} a\\2b \end{bmatrix}$

Now, this means that your equation looks like:
$\begin{bmatrix} a\\2b \end{bmatrix}=\begin{bmatrix} 53 \\ 98 \end{bmatrix}$

This simply means:

a=53

and

2b=98 or b=49

Therefore, a+b=53+49=102

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Example Question #21 : Matrices

$A = \begin{bmatrix} 0&0& 0 & 3\\ 0& 0 & 3& 0\\ 0& 3& 0& 0 \\3& 0&0& 0 \end{bmatrix}$

Evaluate $A^{2}$ .

Possible Answers:

$A ^{2}= \begin{bmatrix} 0&0& 0 & 0 \\ 0&0& 0 & 0 \\ 0&0& 0 & 0\\ 0&0& 0 & 0 \end{bmatrix}$

$A ^{2}= \begin{bmatrix} 9& 0&0& 0\\ 0& 9& 0& 0\\ 0& 0 & 9& 0\\ 0&0& 0 & 9 \end{bmatrix}$

$A ^{2}= \begin{bmatrix}9 & 0&0& 0 \\0&0& 0 & 9 \\ 0& 0 & 9& 0 \\ 0& 9& 0& 0 \end{bmatrix}$

$A ^{2}= \begin{bmatrix}0& 0 & 9& 0 \\ 0& 9& 0& 0 \\ 9 & 0&0& 0 \\0&0& 0 & 9 \end{bmatrix}$

$A ^{2}= \begin{bmatrix}0&0& 0 & 9\\0& 0 & 9& 0 \\ 0& 9& 0& 0 \\ 9 & 0&0& 0 \end{bmatrix}$

Correct answer:

$A ^{2}= \begin{bmatrix} 9& 0&0& 0\\ 0& 9& 0& 0\\ 0& 0 & 9& 0\\ 0&0& 0 & 9 \end{bmatrix}$

Explanation:

$A^{2} = A \cdot A = \begin{bmatrix} 0&0& 0 & 3\\ 0& 0 & 3& 0\\ 0& 3& 0& 0 \\3& 0&0& 0 \end{bmatrix}\begin{bmatrix} 0&0& 0 & 3\\ 0& 0 & 3& 0\\ 0& 3& 0& 0 \\3& 0&0& 0 \end{bmatrix}$

The element in row , column , of $A^{2}$ can be found by multiplying row of by row of - that is, by multiplying elements in corresponding positions and adding the products. Therefore,

$A^{2} = \begin{bmatrix} 0 \cdot 0 + 0 \cdot 0 +0 \cdot 0 +3 \cdot 3 &0 \cdot 0 +0 \cdot 0 + 0 \cdot 3 +3 \cdot 0 &0 \cdot 0 + 0 \cdot 3 +0 \cdot 0 +3 \cdot 0&0 \cdot 3 + 0 \cdot 0 +0 \cdot 0 +3 \cdot 0 \\ 0 \cdot 0 + 0 \cdot 0 +3 \cdot 0 +0 \cdot 3 &0 \cdot 0 +0 \cdot 0 + 3 \cdot 3 +0 \cdot 0 &0 \cdot 0 + 0 \cdot 3 +3 \cdot 0 +0 \cdot 0&0 \cdot 3 + 0 \cdot 0 +3 \cdot 0 +0 \cdot 0\\ 0 \cdot 0 + 3 \cdot 0 +0 \cdot 0 +0 \cdot 3 &0 \cdot 0 +3 \cdot 0 + 0 \cdot 3 +0 \cdot 0 &0 \cdot 0 + 3 \cdot 3 +0 \cdot 0 +0\cdot 0&0 \cdot 3 + 3 \cdot 0 +0 \cdot 0 +0 \cdot 0\\3 \cdot 0 + 0 \cdot 0 +0 \cdot 0 +0 \cdot 3 &3 \cdot 0 +0 \cdot 0 + 0 \cdot 3 +0 \cdot 0 &3 \cdot 0 + 0 \cdot 3 +0 \cdot 0 +0 \cdot 0&3 \cdot 3 + 0 \cdot 0 +0 \cdot 0 +0\cdot 0 \end{bmatrix}$

$= \begin{bmatrix} 0+0+0+9 & 0 +0+0+0 & 0 +0+0+0 & 0 +0+0+0 \\ 0 +0+0+0 & 0+0+9+0 & 0 +0+0+0& 0 +0+0+0 \\ 0 +0+0+0 & 0 +0+0+0& 0+9+0+0 & 0 +0+0+0 \\0 +0+0+0 & 0 +0+0+0& 0 +0+0+0 & 9+0+0+0 \end{bmatrix}$

$= \begin{bmatrix} 9& 0&0& 0\\ 0& 9& 0& 0\\ 0& 0 & 9& 0\\ 0&0& 0 & 9 \end{bmatrix}$

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Example Question #22 : Matrices

$A =\begin{bmatrix} - 3 & x\\ 5& 4 \end{bmatrix}$

The determinant of this matrix is equal to 4. Evaluate .

Possible Answers:

$x = -3 \frac{1}{5}$

$x = 3 \frac{1}{5}$

x = -8

x = 8

x = -2

Correct answer:

$x = -3 \frac{1}{5}$

Explanation:

A matrix $A = \begin{bmatrix} a& b\\ c& d \end{bmatrix}$ has as its determinant D = ad-bc . Setting a= -3, b = x, c = 5, d = 4 , this becomes

$D = ad-bc =-3 \cdot 4 - x \cdot 5 = -12 - 5x$

Set this determinant equal to 4 and solve for :

-12 - 5x = 4

-12 - 5x + 12 = 4 + 12

-5x = 16

$\frac{-5x}{-5 } = \frac{16}{-5}$

$x = -3 \frac{1}{5}$

the correct response.

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All SAT II Math II Resources

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SAT II Math II : SAT Subject Test in Math II

Study concepts, example questions & explanations for SAT II Math II

All SAT II Math II Resources

Example Questions

Example Question #2 : Matrices

Example Question #13 : Find The Sum Or Difference Of Two Matrices

Example Question #1 : Matrices

Example Question #1 : How To Find Scalar Interactions With A Matrix

Example Question #1061 : Algebra

Example Question #4 : Find The Sum Or Difference Of Two Matrices

Example Question #1 : How To Subtract Matrices

Example Question #1062 : Algebra

Example Question #21 : Matrices

Example Question #22 : Matrices

All SAT II Math II Resources

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