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Precalculus : Find the Sum of Two Matrices

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Example Question #1 : Find The Sum Of Two Matrices

Add:

$\begin{bmatrix} 11& 9\\ 7& -5 \end{bmatrix}+$ $\begin{bmatrix} 6& 9\\ 10& 13 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 36\\ 31 \end{bmatrix}$

$\begin{bmatrix} 17& 18\\ 17& 8 \end{bmatrix}$

$\begin{bmatrix} 35\\ 25 \end{bmatrix}$

$\begin{bmatrix} 17& 19\\ 16& 15 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 17& 18\\ 17& 8 \end{bmatrix}$

Explanation:

In order to add matrices, they have to be of the same dimension. In this case, they are both 2x2.

The formula to add matrices is as follows:

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

Plugging in our values to solve we get:

$\begin{bmatrix} 11& 9\\ 7& -5 \end{bmatrix}+\begin{bmatrix} 6& 9\\ 10& 13 \end{bmatrix}=\begin{bmatrix} 11+6& 9+9\\ 7+10& -5+13 \end{bmatrix}=\begin{bmatrix} 17& 18\\ 17& 8 \end{bmatrix}$

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Example Question #2 : Find The Sum Of Two Matrices

Add:

$3\begin{bmatrix} 2& 0\\ 1& 4 \end{bmatrix}+$ $\begin{bmatrix} 2& 7\\ 5& 1 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 12& 21\\ 18& 15 \end{bmatrix}$

$\begin{bmatrix} 4& 7\\ 6& 5 \end{bmatrix}$

$\begin{bmatrix} 24& 21\\ 18& 15 \end{bmatrix}$

$\begin{bmatrix} 8& 7\\ 8& 13 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 8& 7\\ 8& 13 \end{bmatrix}$

Explanation:

The first step in solving this problem is to multiply the matrix by the scalar. The formula is as follows:

$S\begin{bmatrix}a &b\\c &d\end{bmatrix}= \begin{bmatrix} Sa &Sb \\ Sc & Sd\end{bmatrix}$

$3 \begin{bmatrix}2 &0 \\1 &4 \end{bmatrix} =\begin{bmatrix} 3(2)&3(0)\\3(1)& 3(4)\end{bmatrix}= \begin{bmatrix} 6 &0 \\3 &12\end{bmatrix}$

In order to add matrices, they have to be of the same dimension.

In this case, they are both 2x2. So, we can then add the matrices together. The result is as follows:

$\begin{bmatrix} 6 &0\\3 &12\end{bmatrix}+\begin{bmatrix} 2 & 7 \\5 &1 \end{bmatrix}=\begin{bmatrix} 6+2 &0+7\\3+5& 12+1\end{bmatrix}$

$=\begin{bmatrix} 8 &7\\8&13\end{bmatrix}$

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Example Question #3 : Addition Of Matrices

We consider and defined as follows where they are supposed to be of order .

$A=\begin{bmatrix} 1 & 2 & 3 &\cdots & n\\ 1 &2 &3 &\cdots &n \\ 1& 2 &3 &\cdots & n\\ \vdots&\vdots &\vdots &\vdots &\vdots \\ 1& 2 &3 &\cdots &n \end{bmatrix}$ $B=\begin{bmatrix} n&n-1 &n-2 &\cdots &1 \\ n&n-1 &n-2 &\cdots &1\\\vdots & \vdots&\vdots &\vdots &\vdots \\ n&n-1 &n-2 &\cdots &1\\ n&n-1 &n-2 &\cdots &1 \end{bmatrix}$

What is the sum of and ?

Possible Answers:

$\begin{bmatrix}\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} \\ \frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} \\\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} \\ \vdots &\vdtos &\vdots &\vdots &\vdots \\ \frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} &\frac{n+1}{2} \end{bmatrix}$

$\begin{bmatrix}n+2 &n+2 &n+2 &n+2 &n+2 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ \vdots &\vdtos &\vdots &\vdots &\vdots \\ n+1 &n+1 &n+1 &n+1 &n+1 \end{bmatrix}$

$\begin{bmatrix}n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ \vdots &\vdtos &\vdots &\vdots &\vdots \\ n-1 &n-1 &n-1 &n-1 &n-1 \end{bmatrix}$

$\begin{bmatrix}n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ \vdots &\vdtos &\vdots &\vdots &\vdots \\ n+2 &n+2 &n+2 &n+2 &n+2 \end{bmatrix}$

$\begin{bmatrix}n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ \vdots &\vdtos &\vdots &\vdots &\vdots \\ n+1 &n+1 &n+1 &n+1 &n+1 \end{bmatrix}$

Correct answer:

$\begin{bmatrix}n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ \vdots &\vdtos &\vdots &\vdots &\vdots \\ n+1 &n+1 &n+1 &n+1 &n+1 \end{bmatrix}$

Explanation:

We can perform the addition since the matrices have the same sizes.

$A+B=\begin{bmatrix} 1 & 2 & 3 &\cdots & n\\ 1 &2 &3 &\cdots &n \\ 1& 2 &3 &\cdots & n\\ \vdots&\vdots &\vdots &\vdots &\vdots \\ 1& 2 &3 &\cdots &n \end{bmatrix}+$ $\begin{bmatrix} n&n-1 &n-2 &\cdots &1 \\ n&n-1 &n-2 &\cdots &1\\\vdots & \vdots&\vdots &\vdots &\vdots \\ n&n-1 &n-2 &\cdots &1\\ n&n-1 &n-2 &\cdots &1 \end{bmatrix}$

Looking at the first row of entries we get:

$\begin{bmatrix} 1+n & 2+n-1=1+n & 3+n-2=1+n ...\end{bmatrix}$

Note that any entry in the sum of A+B is equal to n+1 .

Thus the sum becomes:

$\begin{bmatrix}n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ n+1 &n+1 &n+1 &n+1 &n+1 \\ \vdots &\vdtos &\vdots &\vdots &\vdots \\ n+1 &n+1 &n+1 &n+1 &n+1 \end{bmatrix}$

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Example Question #4 : Addition Of Matrices

We consider the matrices and of the same size .

$A=\begin{bmatrix} 1 &1 &\cdots &1 \\ 1&1 &\cdots &1 \\ \vdots &\vdots &\vdots &\vdots \\ 1& 1&\cdots &1 \end{bmatrix},B=\begin{bmatrix} 1 &-1 &\cdots &-1 \\ 1&-1 &\cdots &-1 \\ \vdots &\vdots &\vdots &\vdots \\ 1& -1&\cdots &-1 \end{bmatrix}$

Find the sum A+B .

Possible Answers:

$A+B=\begin{bmatrix} 2 &1 &\cdots & 0\\ 1&1 &\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 2& 1&\cdots &1 \end{bmatrix}$

$A+B=\begin{bmatrix} 0&0 &\cdots &0\\ 0&0 &\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 0& 0&\cdots &0 \end{bmatrix}$

$A+B=\begin{bmatrix} 1 &0 &\cdots &-1 \\ 1&0 &\cdots &-1 \\ \vdots &\vdots &\vdots &\vdots \\ 1& 0&\cdots &-1 \end{bmatrix}$

$A+B=\begin{bmatrix} 2 &1 &\cdots &1 \\ 2&1 &\cdots &1 \\ \vdots &\vdots &\vdots &\vdots \\ 2& 1&\cdots &1 \end{bmatrix}$

$A+B=\begin{bmatrix} 2&0&\cdots &0 \\ 2&0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 2& 0&\cdots &0 \end{bmatrix}$

Correct answer:

$A+B=\begin{bmatrix} 2&0&\cdots &0 \\ 2&0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 2& 0&\cdots &0 \end{bmatrix}$

Explanation:

Note: For addition of matrices, we do it componentwise.

Note: Adding the first two columns of and we obtain for every row of the first column of the resulting matrix.

In all other case, we obtain 0 everywhere. This gives the matrix:

$A+B=\begin{bmatrix} 2&0&\cdots &0 \\ 2&0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 2& 0&\cdots &0 \end{bmatrix}$

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Example Question #5 : Addition Of Matrices

We will consider the two matrices and given below. and are of the same size.

$A=\begin{bmatrix} 2&0&\cdots &0 \\ 2&2&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 2& 2&\cdots &2\end{bmatrix},B= \begin{bmatrix} -2&0&\cdots &0 \\ -2&-2&\cdots &0 \\ \cdots &\cdots &\cdots &\cdots \\-2&-2&-2&-2\\\ 2& 2&\cdots &2 \end{bmatrix}$

Find the sum A+B.

Possible Answers:

$A+B=\begin{bmatrix} 0&0&\cdots &0 \\ 0&0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 0& 4&\cdots &4 \end{bmatrix}$

$A+B=\begin{bmatrix} 0& 0 &\cdots & 0 \\ &0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 0& 2&\cdots &0 \end{bmatrix}$

$A+B=\begin{bmatrix} 0&0&\cdots &0 \\ 0&0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 4& 4&\cdots &4 \end{bmatrix}$

$A+B=\begin{bmatrix} 0&0&\cdots &0 \\ 0&0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 0& 0&\cdots &0 \end{bmatrix}$

$A+B=\begin{bmatrix} 0&0&\cdots &0 \\ 0&0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 0& 0&\cdots &4 \end{bmatrix}$

Correct answer:

$A+B=\begin{bmatrix} 0&0&\cdots &0 \\ 0&0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 4& 4&\cdots &4 \end{bmatrix}$

Explanation:

Adding componentwise (adding entry by entry) we obtain zeros everywhere except for the last row where we get 2+2 to obain in every component of the row.

This gives the matrix:

$A+B=\begin{bmatrix} 0&0&\cdots &0 \\ 0&0&\cdots &0 \\ \vdots &\vdots &\vdots &\vdots \\ 4& 4&\cdots &4 \end{bmatrix}$

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Example Question #2 : Find The Sum Of Two Matrices

We consider the two matrice, find the sum A+B .

$A=\begin{bmatrix} 1 &2 & 3 &4 &5 \end{bmatrix},B=\begin{bmatrix} 1 &2 & 3 &4 &5 \end{bmatrix}$

Possible Answers:

$A+B=\begin{bmatrix} 2 &4 & 6 &8 &10 \end{bmatrix}$

$A+B=\begin{bmatrix} 2 &4 & 0 &8 &10 \end{bmatrix}$

$A+B=\begin{bmatrix} 2 &4 & 6 &8 &0 \end{bmatrix}$

We can't add A and B since they are not matrices

$A+B=\begin{bmatrix} 1 &4 & 6 &8 &10 \end{bmatrix}$

Correct answer:

$A+B=\begin{bmatrix} 2 &4 & 6 &8 &10 \end{bmatrix}$

Explanation:

Since A and B have the same size, we can perform addition.

The addition is performed componentwise.

The entry located at (i,j) of matrix A is added to the entry located at (i,j) of the matrix B.

In this case i=1 and j=1, 2 ,3 ,4,5

performing this operation we obtain:

$\begin{bmatrix} 2 &4 & 6 &8 &10 \end{bmatrix}$

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Example Question #151 : Matrices And Vectors

We consider the matrices and given below. Find the sum A+B .

$A=\begin{bmatrix} 1\\1 \\ 1 \\ 1\\ 1 \end{bmatrix}$ , $B=\begin{bmatrix} 1\\1 \\ 1 \\ 1\\ 1 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 1\\1 \\ 2 \\ 1\\ 1 \end{bmatrix}$

$\begin{bmatrix} 2\\2 \\ 2 \\ 2\\ 2 \end{bmatrix}$

We can't perform this addition.

$\begin{bmatrix} 2\\1 \\ 1 \\ 1\\ 1 \end{bmatrix}$

$\begin{bmatrix} 1\\1 \\ 1 \\ 1\\ 1 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 2\\2 \\ 2 \\ 2\\ 2 \end{bmatrix}$

Explanation:

Since A and B have the same size, we can perform addition.

The addition is performed componentwise.

The entry located at (i,j) of matrix A is added to the entry located at (i,j) of the matrix B.

Performing this operation we obtain:

$\begin{bmatrix} 2 \\2 \\2 \\2 \\2 \end{bmatrix}$

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Example Question #8 : Addition Of Matrices

For the matrices below, find the sum A+B ( and are assumed to have the same size) . is assumed to be an odd positive integer.

$A=\begin{bmatrix}1 &1 &1 &1 \\ 1&1 & 1 &1 \\ \vdots&\vdots&\vdots&\vdots \\ 1& 1 &1 &1 \end{bmatrix}$

$B=\begin{bmatrix} (-1)^m& (-1)^m & (-1)^m & (-1)^m \\ \vdots&\vdots &\vdots &\vodts \\ (-1)^m & (-1)^m & (-1)^m & (-1)^m \\ (-1)^m& (-1)^m & (-1)^m & (-1)^m \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 1 &1 &1 & 1\\ 1&1 &1 &1\\\vdots&\vdots&\vdots&\vdots \\ -2 &-2 &-2 &-2 \end{bmatrix}$

$\begin{bmatrix} 0 &0 &0 &0 \\ 0&0 &0 &0 \\ \vdots&\vdots&\vdots&\vdots\\ 0& 0 &0 &0 \end{bmatrix}$

$\begin{bmatrix} -2 &-2 &-2 &-2 \\ -2 &-2 &-2 &-2 \\ \vdots& \vdots& \vdots& \vdots\\-2 &-2 &-2 &-2 \end{bmatrix}$

$\begin{bmatrix} 2 &2 &2 &2 \\ 2&2 & 2&2 \\ \vdots& \vdots& \vdots& \vdots\\ 2 &2 & 2 &2 \end{bmatrix}$

$\begin{bmatrix} 1 &1 & 1 &1 \\ 1 &1 & 1 &1 \\ \vdots& \vdots& \vdots& \vdots \\ 1 &1 & 1 &1 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 0 &0 &0 &0 \\ 0&0 &0 &0 \\ \vdots&\vdots&\vdots&\vdots\\ 0& 0 &0 &0 \end{bmatrix}$

Explanation:

Since we are assuming that the two matrices have the same size, we can performthe matrices addition.

We know that when adding matrices, we add them componenwise. Let (i,j) be any entry of the addition matrix. We add the entry from A to the entry from B:

since the entries from A are the same and given by 1 and the entries from B are the same and given by (-1)^m , we add these two to obtain :

1+(-1)^m and we know that m is odd integer, hence (-1)^m=-1 .

Therefore the entry of the sum matrix is 0

Therefore our matrix is given by:

$\begin{bmatrix} 0 &0 &0 &0 \\ 0&0 &0 &0 \\ \vdots&\vdots&\vdots&\vdots\\ 0& 0 &0 &0 \end{bmatrix}$

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Example Question #9 : Addition Of Matrices

Simplify:

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

Possible Answers:

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

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

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

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

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

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 #10 : Addition Of Matrices

Simplify:

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

Possible Answers:

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

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

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

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

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

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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Precalculus : Find the Sum of Two Matrices

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Sum Of Two Matrices

Example Question #2 : Find The Sum Of Two Matrices

Example Question #3 : Addition Of Matrices

Example Question #4 : Addition Of Matrices

Example Question #5 : Addition Of Matrices

Example Question #2 : Find The Sum Of Two Matrices

Example Question #151 : Matrices And Vectors

Example Question #8 : Addition Of Matrices

Example Question #9 : Addition Of Matrices

Example Question #10 : Addition Of Matrices

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