Pre-Calculus - Find the Sum or Difference of Two Matrices

Let

$A=\begin{pmatrix} 5&1 \ 3&-7 \end{pmatrix}$ $B=\begin{pmatrix} 4&2 \ 9& 0 \end{pmatrix}$

Determine the sum .

$\begin{pmatrix} 9&3 \ 12&-7 \end{pmatrix}$

DNE

$\begin{pmatrix} 1&-1 \ -6&7 \end{pmatrix}$

$\begin{pmatrix} 4&2 \ 9&-7 \end{pmatrix}$

Explanation

Since the dimensions of the two matrices are equal the sum of the two matrices exists.

To find the sum, add each component entry from the first matrix to the same component entry of the second matrix.

$A+B=\begin{pmatrix} 5&1 \ 3&-7 \end{pmatrix}+\begin{pmatrix} 4&2 \ 9& 0 \end{pmatrix}}=\begin{pmatrix} 9&3 \ 12&-7 \end{pmatrix}$

We consider the following matrices:

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

Find .

We can't find .

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

$\begin{bmatrix} 0 & 0 &2 \0 & 0 & 2\0 &0 &2 \end{bmatrix}$

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

$\begin{bmatrix} 0 &0 &0 &2 \0 & 0 &0 &2 \ 0&0 &0 &2 \ 0 &0 &0 &2 \end{bmatrix}$

Explanation

To be able to subtract matrices, the matrices must have the same size.

A is 3x3 and B is 3x4. Therefore we can't perform this operation.

Add:

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

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

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

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

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

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:

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

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 ?

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,

$\begin{bmatrix} 7& 16& -3\ -13& 5&8 \ 4&-2 &21 \end{bmatrix}+\begin{bmatrix} -11& 13& 4\ 6& 5& -18\ 16& 3& -9 \end{bmatrix}=$

$\begin{bmatrix} -4& 29& 1\ -7& 10& -10\ 20& 1& 12 \end{bmatrix}$

$\begin{bmatrix} -4& 29& 1\ -9& 10& -11\ 21& 11& 12 \end{bmatrix}$

$\begin{bmatrix} -4& 29& 1\ -7& 8& -10\ 20& -3& 12 \end{bmatrix}$

$\begin{bmatrix} -4& -29& -1\ -7& -5& -1\ 20& 8& 12 \end{bmatrix}$

$\begin{bmatrix} 4& 29& -1\ -17& 10& 10\ 20& -1& -12 \end{bmatrix}$

Explanation

To find the sum of two matrices, we simply add each entry from one matrix to the corresponding entry of the other matrix, and the result becomes the entry in the same location of the matrix for their sum:

$\begin{bmatrix} 7& 16& -3\ -13& 5&8 \ 4&-2 &21 \end{bmatrix}+\begin{bmatrix} -11& 13& 4\ 6& 5& -18\ 16& 3& -9 \end{bmatrix}=\begin{bmatrix} -4& 29& 1\ -7& 10& -10\ 20& 1& 12 \end{bmatrix}$

Find the sum of the two matrices

$\bigl(\begin{smallmatrix} 1 \end{smallmatrix}\bigr) +\bigl(\begin{smallmatrix} 2&2 \end{smallmatrix}\bigr)$

$\bigl(\begin{smallmatrix} 3 \end{smallmatrix}\bigr)$

$\bigl(\begin{smallmatrix} 3&3 \end{smallmatrix}\bigr)$

$\bigl(\begin{smallmatrix} 2 \end{smallmatrix}\bigr)$

Explanation

Because the matrices do not have the same dimension, the sum does not exist.

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

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

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

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

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

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

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

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

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 , we add these two to obtain :

and we know that m is odd integer, hence .

Therefore the entry of the sum matrix is 0

Therefore our matrix is given by:

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

Subtract:

$\begin{bmatrix} 2& 6\ 5& 3 \end{bmatrix}-\begin{bmatrix} 1& 7 \ 9& 2 \end{bmatrix}$

$\begin{bmatrix} 1& -1\ -4& 1 \end{bmatrix}$

$\begin{bmatrix} 1& -3\ -2& 1 \end{bmatrix}$

$\begin{bmatrix} 52& 2\ -22& 29 \end{bmatrix}$

$\begin{bmatrix} 0\ -3 \end{bmatrix}$

Explanation

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

The formula to subtract matrices is as follows:

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

Using this formula we plug in our values and solve:

$\begin{bmatrix} 1 &3 \2 &-1 \end{bmatrix}+\begin{bmatrix} 0 &5 \ -2&6 \end{bmatrix}$

$\begin{bmatrix} 1 &8 \ 0&5 \end{bmatrix}$

$\begin{bmatrix} 1 &8 \ 9&5 \end{bmatrix}$

$\begin{bmatrix} 1 &5 \ 0&5 \end{bmatrix}$

$\begin{bmatrix} -1 &8 \ 0&5 \end{bmatrix}$

$\begin{bmatrix} 1 &8 \ 0&2 \end{bmatrix}$

Explanation

To add two matrices, combine the corresponding numbers in the matrices: (). Then, put those answers in the corresponding spots in the answer matrix. Therefore, your answer is $\begin{bmatrix} 1 &8 \ 0&5 \end{bmatrix}$ .

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 .

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

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

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

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

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

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:

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

Find the Sum or Difference of Two Matrices

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