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ACT Math : How to find an answer with a matrix

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Example Questions

Example Question #1 : Other Matrices

With matrix notation, what does M_2x3 x N_3x4 equal?

Possible Answers:

None of the answers are correct

P_2x4

P_3x4

P_3x3

P_4x2

Correct answer:

P_2x4

Explanation:

M_2x3 x N_3x4 = P_2x4

In general matrix notation, M_rxc shows that the matrix is named M and r is the number of rows and c is the number of columns. When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. In addition, when adding or subtracting matrices, the matrices must be of the same size.

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Example Question #2 : Other Matrices

What is the solution to the following matrix?

$\begin{vmatrix}8 & 3\\ -2 & 4 \end{vmatrix}$

Possible Answers:

Correct answer:

Explanation:

In order to solve the matrix, the determinant rule "ad-bc" must be used. is in the "a" position, is in the "b" position, is in the "c" position, and is in the "d" position. After plugging the numbers into "ad-bc," we get

$(8\times 4)-(-2\times 3)=(32)-(-6)=32+6=38$

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Example Question #3 : Other Matrices

Which of the following augmented matrices can be used to solve this system of equations?

y = 5x

x+ y = 3z

2x - 3y + 4z = 100

Possible Answers:

$\begin{bmatrix} 5 & -1 & 0 \\ 1 & 1 & -3 \\2 & - 3 & 4 \end{matrix} \begin{vmatrix} 0 \\0\\ 100 \end{bmatrix}$

$\begin{bmatrix} 5 & -1 & 0 \\ 1 & 1 & 3 \\2 & - 3 & 4 \end{matrix} \begin{vmatrix} 0 \\0\\ 100 \end{bmatrix}$

$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & 0 \\2 & - 3 & 4 \end{matrix} \begin{vmatrix} 5 \\3\\ 100 \end{bmatrix}$

$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & -3 \\2 & - 3 & 4 \end{matrix} \begin{vmatrix} 5 \\0\\ 100 \end{bmatrix}$

$\begin{bmatrix} 5 & -1 & 0 \\ 1 & 1 & 0 \\2 & - 3 & 4 \end{matrix} \begin{vmatrix} 0 \\3\\ 100 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 5 & -1 & 0 \\ 1 & 1 & -3 \\2 & - 3 & 4 \end{matrix} \begin{vmatrix} 0 \\0\\ 100 \end{bmatrix}$

Explanation:

To set up and augmented matrix for a 3x3 system of equations, all equations must be in standard form Ax+By + Cz= D . The third equation is already in standard form; the first two are not and must be rewritten as such.

y = 5x

y - y = 5x - y

5x - y = 0

x+ y = 3z

x+ y- 3z = 3z - 3z

x+ y- 3z = 0

The system is now

5x - y = 0

x+ y- 3z = 0

2x - 3y + 4z = 100

Write the augmented matrix with each row comprising the coefficients of one equation in order:

$\begin{bmatrix} 5 & -1 & 0 \\ 1 & 1 & -3 \\2 & - 3 & 4 \end{matrix} \begin{vmatrix} 0 \\0\\ 100 \end{bmatrix}$

is the correct choice.

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Example Question #4 : Other Matrices

Which of the following augmented matrices can be used to solve this system of equations?

y = 4x + 7

2x - 7y = 8

Possible Answers:

$\begin{bmatrix} 1&-4 \\ 2& -7 \end{matrix} \begin{vmatrix} 7\\ 8 \end{bmatrix}$

$\begin{bmatrix} 4& 1 \\ 2& -7 \end{matrix} \begin{vmatrix} 7\\ 8 \end{bmatrix}$

$\begin{bmatrix} 1&4 \\ 2& -7 \end{matrix} \begin{vmatrix} 7\\ 8 \end{bmatrix}$

$\begin{bmatrix} 4&-1 \\ 2& -7 \end{matrix} \begin{vmatrix} 7\\ 8 \end{bmatrix}$

$\begin{bmatrix} 4&-1 \\ 2& -7 \end{matrix} \begin{vmatrix} -7\\ 8 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 4&-1 \\ 2& -7 \end{matrix} \begin{vmatrix} -7\\ 8 \end{bmatrix}$

Explanation:

To set up and augmented matrix for a 2x2 system of equations, both equations must be in standard form Ax+By = C . The second equation is already in standard form.

Rewrite the first equation in standard form as follows:

y = 4x + 7

4x + 7 = y

4x + 7- y - 7 = y - y - 7

4x - y = - 7

The system has been rewritten as

4x - y = - 7

2x - 7y = 8

Write the augmented matrix with each row comprising the coefficients of one equation in order:

$\begin{bmatrix} 4&-1 \\ 2& -7 \end{matrix} \begin{vmatrix} -7\\ 8 \end{bmatrix}$

is the correct choice.

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Example Question #5 : Other Matrices

Read the following question:

A high school choir sold large boxes of cookies for $5.75 each, medium boxes for $4.75 each, and small boxes for $3.25 each. The band sold a total of 445 boxes and raised a total of $1,924.25. There were twenty more medium boxes sold than large boxes.

Which of the following augmented matrices represents the system of equations that could be set up to solve this problem?

Possible Answers:

$\begin{bmatrix} 1 & 1 & 5.75 \\ 1 &-1 & 4.75 \\0 & 1 & 3.25 \end{matrix} \begin{vmatrix} 20 \\445\\ 1,924.25\end{bmatrix}$

$\begin{bmatrix} 1 & -1 & 5.75 \\ 1 &-1 & 4.75 \\0 & 1 & 3.25 \end{matrix} \begin{vmatrix} -20 \\445\\ 1,924.25\end{bmatrix}$

$\begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\5.75 & 4.75 & 3.25 \end{matrix} \begin{vmatrix} 20 \\445\\ 1,924.25\end{bmatrix}$

$\begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & 1 \\5.75 & 4.75 & 3.25 \end{matrix} \begin{vmatrix} -20 \\445\\ 1,924.25\end{bmatrix}$

$\begin{bmatrix} 1 & -1 & 5.75 \\ 1 &-1 & 4.75 \\0 & 1 & 3.25 \end{matrix} \begin{vmatrix} 20 \\445\\ 1,924.25\end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & 1 \\5.75 & 4.75 & 3.25 \end{matrix} \begin{vmatrix} -20 \\445\\ 1,924.25\end{bmatrix}$

Explanation:

If we let , , and represent the number of large, medium, and small boxes sold, respectively, since twenty more medium boxes than large were sold, one equation of the 3x3 system will be

y = x+20

or, in standard form,

x-y = -20

Since 445 boxes were sold, another linear equation will be

x+y+z = 445

The money raised from the sale of large boxes of cookies, each of which cost $5.75, is 5.75x ; the money raised from the sale of small boxes of cookies, each of which cost $4.75, is 4.75y ; and the money raised from the sale of small boxes of cookies, each of which cost $3.25, is 3.25z .

The total money raised is $1,924.25, so the other linear equation of the system is

5.75x+ 4.75y+ 3.25z= 1,924.25

The augmented matrix of this system will comprise the coefficients of these equations, all of which are now standard form, so the matrix will be

$\begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & 1 \\5.75 & 4.75 & 3.25 \end{matrix} \begin{vmatrix} -20 \\445\\ 1,924.25\end{bmatrix}$ ,

which is the correct choice.

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Example Question #6 : Other Matrices

Read the following question:

A high school band sold large boxes of cookies for $4.75 each and small boxes of cookies for $3.25 each. The band sold a total of 305 boxes and raised a total of $1,196.75.

Which of the following augmented matrices represents the system of equations that could be set up to solve this problem?

Possible Answers:

$\begin{bmatrix} 1&1 \\ 4.75&3.25 \end{matrix} \begin{vmatrix}1,196.75 \\ 305 \end{bmatrix}$

$\begin{bmatrix}3.25&4.75 \\ 305 &1,196.75 \end{matrix} \begin{vmatrix}1 \\ 1 \end{bmatrix}$

$\begin{bmatrix} 1&4.75 \\ 1&3.25 \end{matrix} \begin{vmatrix}1,196.75 \\ 305 \end{bmatrix}$

$\begin{bmatrix} 1&4.75 \\ 1&3.25 \end{matrix} \begin{vmatrix} 305 \\ 1,196.75 \end{bmatrix}$

$\begin{bmatrix} 1&1 \\ 4.75&3.25 \end{matrix} \begin{vmatrix} 305 \\ 1,196.75 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1&1 \\ 4.75&3.25 \end{matrix} \begin{vmatrix} 305 \\ 1,196.75 \end{bmatrix}$

Explanation:

If we let and represent the number of large and small boxes sold, respectively, since 305 boxes were sold, one linear equation of the 2x2 system will be

x+y = 305

The money raised from the sale of large boxes of cookies, each of which cost $4.75, is 4.75x ; the money raised from the sale of small boxes of cookies, each of which cost $3.25, is 3.25y . The total money raised is $1,196.75, so the other linear equation of the system is

4.75x+ 3.25y=1,196.75

The augmented matrix of this system will comprise the coefficients of these equations, both of which are in standard form, so the matrix will be

$\begin{bmatrix} 1&1 \\ 4.75&3.25 \end{matrix} \begin{vmatrix} 305 \\ 1,196.75 \end{bmatrix}$

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Example Question #7 : Other Matrices

Read the following question:

A chemist needs one liter of a solution of 20% alcohol for an experiment. However, he only has two solutions on hand, one of which is 10% alcohol and one of which is 40% alcohol. How much of each solution must he mix in order to make his desired solution?

Which of the following augmented matrices represents the system of equations that could be set up to solve this problem?

Possible Answers:

$\begin{bmatrix} 1&0.1 \\ 1 &0.4 \end{matrix} \begin{vmatrix} 1 \\2 \end{bmatrix}$

$\begin{bmatrix} 1&1 \\ 0.1 &0.4 \end{matrix} \begin{vmatrix} 1 \\0.2 \end{bmatrix}$

$\begin{bmatrix} 1&1 \\ 0.1 &0.4 \end{matrix} \begin{vmatrix} 1 \\2 \end{bmatrix}$

$\begin{bmatrix} 1&0.1 \\ 1 &0.4 \end{matrix} \begin{vmatrix} 2 \\1 \end{bmatrix}$

$\begin{bmatrix} 1&1 \\ 0.1 &0.4 \end{matrix} \begin{vmatrix}0.2 \\1 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1&1 \\ 0.1 &0.4 \end{matrix} \begin{vmatrix} 1 \\0.2 \end{bmatrix}$

Explanation:

If we let and represent the amount of the weaker and stronger solutions, respectively, since the chemist needs 1 liter of the resulting solution, one linear equation of the 2x2 system will be

x+y = 1

The amount of alcohol in liters of a 10% solution will be 0.1x ; the amount of alcohol in liters of a 40% solution will be 0.4y ; and the total amount of alcohol in the resulting solution will be 20 % of a liter, or 0.20 liters. Therefore, the second linear equation of the system will be

0.1x + 0.4y = 0.2

The augmented matrix of this system will comprise the coefficients of these equations, both of which are in standard form, so the matrix will be

$\begin{bmatrix} 1&1 \\ 0.1 &0.4 \end{matrix} \begin{vmatrix} 1 \\0.2 \end{bmatrix}$ ,

which is the correct choice.

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Example Question #1 : Other Matrices

Below is a matrix of the items in Jon's wardrobe:

$\begin{bmatrix} & \text{blue} & \text{red} & \text{white} \\ \text{shirts} & 5 & 3 & 4 \\ \text{pants} &2 &1 &1 \end{bmatrix}$

How many blue items does Jon own?

Possible Answers:

Correct answer:

Explanation:

To find the number of blue items Jon has, we sum the entries in the column labelled $\text{blue}$ and get 2+5=7 . Recall that matrices are organized by rows and columns where each entry refers to the number of items that are, in this case, both of the same color AND type. All the entries in any one column are of the same color. All of the entries in any row are of the same clothing type.

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Example Question #9 : Other Matrices

Below is a matrix of the items in Jon's wardrobe:

$\begin{bmatrix} & \text{blue} & \text{red} & \text{white} \\ \text{shirts} & 5 & 3 & 4 \\ \text{pants} &2 &1 &1 \end{bmatrix}$

How many combinations of blue pants and red shirts can Jon wear for the upcoming 4th of July party?

Possible Answers:

Correct answer:

Explanation:

If Jon has three red shirts and two blue pants. The total number of combinations of one shirt and one pair of pants he can wear is going to be the number of shirts he can wear with the first pair of pants, , plus the number of shirt he can wear with the second pair of pants, also . That sums to total outfits.

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Example Question #10 : Other Matrices

Read the following problem:

The barista at the Teahouse of the December Sun has a problem. He needs to mix twenty pounds of two different kinds of tea together to create a blend called Strawberry Peppermint Delight. The two varieties are Peppermint Nirvana, which costs $12 a pound, and Strawberry Fields, which costs $15 a pound; the new tea will cost $13 a pound, and it will sell for the same price as the two blended teas would separately. How much of each variety will go into the twenty pounds of Strawberry Peppermint Delight?

Which of the following augmented matrices represents the system of equations that could be set up to solve this problem?

Possible Answers:

$\begin{bmatrix} 1&1 \\ 12 & 15 \end{matrix} \begin{vmatrix} 13 \\ 260 \end{bmatrix}$

$\begin{bmatrix} 1&1 \\ 12 & 15 \end{matrix} \begin{vmatrix} 260 \\ 20\end{bmatrix}$

$\begin{bmatrix} 1&1 \\ 12 & 15 \end{matrix} \begin{vmatrix} 20 \\ 13\end{bmatrix}$

$\begin{bmatrix} 1&1 \\ 12 & 15 \end{matrix} \begin{vmatrix} 20 \\ 260 \end{bmatrix}$

$\begin{bmatrix} 1&1 \\ 12 & 15 \end{matrix} \begin{vmatrix} 260 \\ 13\end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1&1 \\ 12 & 15 \end{matrix} \begin{vmatrix} 20 \\ 260 \end{bmatrix}$

Explanation:

If the barista mixes pounds of Peppermint Nirvana and pounds of Strawberry Fields to make twenty pounds of tea total, then

x+y = 20

will be one of the equations in the system.

pounds of Peppermint Nirvana tea for $12 a pound will cost a total of 12x dollars; pounds of Strawberry Fields tea will cost a total of 15y dollars. Tewnty pounds of the Strawberry Peppermint Delight tea for $13 a pound will cost $13 \times 20 = 260$ dollars. Since the tea will sell for the same price blended as separate, the other equation of the system will be

12x + 15y = 260

The augmented matrix of this system will comprise the coefficients of these equations, both of which are in standard form, so the matrix will be

$\begin{bmatrix} 1&1 \\ 12 & 15 \end{matrix} \begin{vmatrix} 20 \\ 260 \end{bmatrix}$ .

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ACT Math : How to find an answer with a matrix

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Other Matrices

Example Question #2 : Other Matrices

Example Question #3 : Other Matrices

Example Question #4 : Other Matrices

Example Question #5 : Other Matrices

Example Question #6 : Other Matrices

Example Question #7 : Other Matrices

Example Question #1 : Other Matrices

Example Question #9 : Other Matrices

Example Question #10 : Other Matrices

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