Matrices - ACT Math
Card 0 of 20
Simplify:
Simplify:
Matrix addition is very easy! All that you need to do is add each correlative member to each other. Think of it like this:
Now, just simplify:
There is your answer!
Matrix addition is very easy! All that you need to do is add each correlative member to each other. Think of it like this:
Now, just simplify:
There is your answer!
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Simplify:
Simplify:
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:
Then, just simplify all of those simple additions and subtractions:
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:
Then, just simplify all of those simple additions and subtractions:
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What is the value of
?
What is the value of
To add matrices you simply add the numbers in the same position as each other.
Plugging the given values into the above formula, we are able to solve the question.
To add matrices you simply add the numbers in the same position as each other.
Plugging the given values into the above formula, we are able to solve the question.
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With matrix notation, what does M2x3 x N3x4 equal?
With matrix notation, what does M2x3 x N3x4 equal?
M2x3 x N3x4 = P2x4
In general matrix notation, Mrxc 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.
M2x3 x N3x4 = P2x4
In general matrix notation, Mrxc 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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What is the solution to the following matrix?
What is the solution to the following matrix?
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
In order to solve the matrix, the determinant rule "ad-bc" must be used.
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Which of the following augmented matrices can be used to solve this system of equations?
Which of the following augmented matrices can be used to solve this system of equations?
To set up and augmented matrix for a 3x3 system of equations, all equations must be in standard form 
. The third equation is already in standard form; the first two are not and must be rewritten as such.
The system is now
Write the augmented matrix with each row comprising the coefficients of one equation in order:
is the correct choice.
To set up and augmented matrix for a 3x3 system of equations, all equations must be in standard form
The system is now
Write the augmented matrix with each row comprising the coefficients of one equation in order:
is the correct choice.
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Which of the following augmented matrices can be used to solve this system of equations?
Which of the following augmented matrices can be used to solve this system of equations?
To set up and augmented matrix for a 2x2 system of equations, both equations must be in standard form 
. The second equation is already in standard form.
Rewrite the first equation in standard form as follows:
The system has been rewritten as
Write the augmented matrix with each row comprising the coefficients of one equation in order:
is the correct choice.
To set up and augmented matrix for a 2x2 system of equations, both equations must be in standard form
Rewrite the first equation in standard form as follows:
The system has been rewritten as
Write the augmented matrix with each row comprising the coefficients of one equation in order:
is the correct choice.
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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?
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?
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
The money raised from the sale of 
large boxes of cookies, each of which cost $4.75, is 
; the money raised from the sale of 
small boxes of cookies, each of which cost $3.25, is 
. The total money raised is $1,196.75, so the other linear equation of the system is
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
If we let
The money raised from the sale of
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
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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?
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?
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
The amount of alcohol in 
liters of a 10% solution will be 
; the amount of alcohol in 
liters of a 40% solution will be 
; 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
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
,
which is the correct choice.
If we let
The amount of alcohol in
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
which is the correct choice.
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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?
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?
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
or, in standard form,
Since 445 boxes were sold, another linear equation will be
The money raised from the sale of 
large boxes of cookies, each of which cost $5.75, is 
; the money raised from the sale of 
small boxes of cookies, each of which cost $4.75, is 
; and the money raised from the sale of 
small boxes of cookies, each of which cost $3.25, is 
.
The total money raised is $1,924.25, so the other linear equation of the system is
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
,
which is the correct choice.
If we let
or, in standard form,
Since 445 boxes were sold, another linear equation will be
The money raised from the sale of
The total money raised is $1,924.25, so the other linear equation of the system is
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
which is the correct choice.
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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?
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?
If the barista mixes 
pounds of Peppermint Nirvana and 
pounds of Strawberry Fields to make twenty pounds of tea total, then
will be one of the equations in the system.
pounds of Peppermint Nirvana tea for $12 a pound will cost a total of 
dollars; 
pounds of Strawberry Fields tea will cost a total of 
dollars. Tewnty pounds of the Strawberry Peppermint Delight tea for $13 a pound will cost 
dollars. Since the tea will sell for the same price blended as separate, the other equation of the system will be
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
.
If the barista mixes
will be one of the equations in the system.
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
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Below is a matrix of the items in Jon's wardrobe:
How many blue items does Jon own?
Below is a matrix of the items in Jon's wardrobe:
How many blue items does Jon own?
To find the number of blue items Jon has, we sum the entries in the column labelled 
and get 
. 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.
To find the number of blue items Jon has, we sum the entries in the column labelled
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Below is a matrix of the items in Jon's wardrobe:
How many combinations of blue pants and red shirts can Jon wear for the upcoming 4th of July party?
Below is a matrix of the items in Jon's wardrobe:
How many combinations of blue pants and red shirts can Jon wear for the upcoming 4th of July party?
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.
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,
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Matthew, Emily, Cynthia, John, and Stephanie are all running for student council. If three equivalent positions are available in this year's election, how many different leadership boards are possible?
Matthew, Emily, Cynthia, John, and Stephanie are all running for student council. If three equivalent positions are available in this year's election, how many different leadership boards are possible?
One possible approach is to enumerate all possibilities using initials for names (ie. MEC, MEJ, MES, etc). If we recognize that order does not matter in this problem, we can also enter 
into our calculators, and that will give us the answer of 
.
One possible approach is to enumerate all possibilities using initials for names (ie. MEC, MEJ, MES, etc). If we recognize that order does not matter in this problem, we can also enter
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Simplify.
Simplify.
When we are asked to simplify a matrix that is being multiplied by a constant we simply multiply each component of the matrix with the scalar factor that is on the outside of the matrix.
Therefore,
When we are asked to simplify a matrix that is being multiplied by a constant we simply multiply each component of the matrix with the scalar factor that is on the outside of the matrix.
Therefore,
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At a sale, a necklace that was originally 
is marked ten percent off. Cara has a coupon that will allow her to get ten percent off the discounted price. How much will Cara pay for the necklace if she uses the coupon?
At a sale, a necklace that was originally
Taking ten percent off of 
gives us 
,
.
Taking ten percent off of this amount gives us 
.
NOTE: This problem cannot be solved by taking twenty percent off of 
.
Taking ten percent off of
Taking ten percent off of this amount gives us
NOTE: This problem cannot be solved by taking twenty percent off of
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Evaluate: 
Evaluate:
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.
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.
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Simplify:
Simplify:
Scalar multiplication and addition of matrices are both very easy. Just like regular scalar values, you do multiplication first:
The addition of matrices is very easy. You merely need to add them directly together, correlating the spaces directly.
Scalar multiplication and addition of matrices are both very easy. Just like regular scalar values, you do multiplication first:
The addition of matrices is very easy. You merely need to add them directly together, correlating the spaces directly.
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What is 
?
What is
You can begin by treating this equation just like it was:
That is, you can divide both sides by 
:
Now, for scalar multiplication of matrices, you merely need to multiply the scalar by each component:
Then, simplify:
Therefore, 
You can begin by treating this equation just like it was:
That is, you can divide both sides by
Now, for scalar multiplication of matrices, you merely need to multiply the scalar by each component:
Then, simplify:
Therefore,
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If 
, what is 
?
If
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 
:
Now, this means that your equation looks like:
This simply means:
and
or 
Therefore, 
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
Now, this means that your equation looks like:
This simply means:
and
Therefore,
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