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Precalculus : Sum or Difference of Two Matrices

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Example Question #1 : Find The Sum Or Difference Of Two Matrices

Subtract:

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

Possible Answers:

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

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

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

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

Correct answer:

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

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:

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Example Question #2 : Find The Sum Or Difference Of Two Matrices

Subtract:

$\begin{bmatrix} 3& 8\\ 10& 12 \end{bmatrix}$ $-\begin{bmatrix} 11\\ 9 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} -61\\ -42 \end{bmatrix}$

$\begin{bmatrix} -33& -72\\ -110& -108 \end{bmatrix}$

No Solution

$\begin{bmatrix} -39\\ 2 \end{bmatrix}$

Correct answer:

No Solution

Explanation:

In order to subtract matrices, they have to be of the same dimension. In this case, they are not. One is 2x2 while the other is 2x1.

As such, we cannot find their difference.

The answer is No Solution.

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Example Question #3 : Find The Sum Or Difference Of Two Matrices

Subtract

$\begin{bmatrix} 10& 5\\ 7& 3 \end{bmatrix}-$ $2\begin{bmatrix} 1& 0\\ 5& 2 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} -8& 5\\ 3& -7 \end{bmatrix}$

$\begin{bmatrix} 8& 5\\ -3& 1 \end{bmatrix}$

$\begin{bmatrix} 8& 5\\ -3& -1 \end{bmatrix}$

$\begin{bmatrix} 12& 5\\ 17& 7 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 8& 5\\ -3& -1 \end{bmatrix}$

Explanation:

The first step in solving this problem is to multiply the matrix by the scalar.

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

$2\begin{bmatrix} 1&0\\5&2\end{bmatrix}=\begin{bmatrix} 2&0\\10&4\end{bmatrix}$

We then have to make sure we can subtract the matrices.

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

Using our new matrix from multiplying the scalar and subtracting it from our other matrix we get the following:

$\begin{bmatrix} 10 &5\\7 &3\end{bmatrix}-\begin{bmatrix} 2&0\\10&4\end{bmatrix}=\begin{bmatrix} 8&5\\-3 & -1\end{bmatrix}$

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Example Question #3 : Find The Sum Or Difference Of Two Matrices

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 B-A .

Possible Answers:

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

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

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

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

We can't find B-A .

Correct answer:

We can't find B-A .

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.

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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 : Find The Sum Or Difference Of Two 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 #7 : Find The Sum Or Difference Of Two Matrices

$\begin{bmatrix} -4&7 &-13 \\ 8&2 &1 \\ -11&-5 &3 \end{bmatrix}-\begin{bmatrix} -6&18 &2 \\ 1&-4 &10 \\ 9&-17 &1 \end{bmatrix}=$

Possible Answers:

$\begin{bmatrix} 10& -1& -5\\ 8& 6& -12\\ -14& 12& 4 \end{bmatrix}$

$\begin{bmatrix} -2& -4& -15\\ 7& -6& -9\\ -20& 6& 2 \end{bmatrix}$

$\begin{bmatrix} -10& -11& 13\\ -3& 4& -9\\ -2& 12& 2 \end{bmatrix}$

$\begin{bmatrix} 2& -13& -8\\ 7& 4& -9\\ -20& 12& -2 \end{bmatrix}$

$\begin{bmatrix} 2& -11& -15\\ 7& 6& -9\\ -20& 12& 2 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 2& -11& -15\\ 7& 6& -9\\ -20& 12& 2 \end{bmatrix}$

Explanation:

To find the difference between two matrices with the same dimensions, we simply subtract each entry from the right matrix from the corresponding entry of the left matrix. After performing the operation 9 times, we get the following matrix:

$\begin{bmatrix} -4&7 &-13 \\ 8&2 &1 \\ -11&-5 &3 \end{bmatrix}-\begin{bmatrix} -6&18 &2 \\ 1&-4 &10 \\ 9&-17 &1 \end{bmatrix}=\begin{bmatrix} 2& -11& -15\\ 7& 6& -9\\ -20& 12& 2 \end{bmatrix}$

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Example Question #5 : Find The Sum Or Difference Of Two Matrices

Find the difference if it exists

$\small \begin{pmatrix} 1\\ 2\\3 \end{pmatrix}-\begin{pmatrix} 2\\2 \\ 2 \end{pmatrix} \,$

Possible Answers:

$\small DNE$

$\small \begin{pmatrix} 1\\ 0\\-1 \end{pmatrix}\,$

$\small \begin{pmatrix} 2\\ 4\\6 \end{pmatrix}$

$\begin{pmatrix} -1\\ 0\\1 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} -1\\ 0\\1 \end{pmatrix}$

Explanation:

Because the two matrices have the same dimension, the difference exists

$\small \small \small \begin{pmatrix} 1\\ 2\\3 \end{pmatrix}-\begin{pmatrix} 2\\2 \\ 2 \end{pmatrix} \,=\begin{pmatrix} 1-2\\ 2-2\\3-2 \end{pmatrix}= \begin{pmatrix} -1\\ 0\\1 \end{pmatrix}$

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Example Question #1 : Find The Sum Or Difference Of Two Matrices

$\begin{bmatrix} 9 &-2 \\ 4&-5 \end{bmatrix}-\begin{bmatrix} 4 &5 \\ -6&-3 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} -5 &-7 \\ 10&-2 \end{bmatrix}$

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

$\begin{bmatrix} 5 &-7 \\ 10&-2 \end{bmatrix}$

$\begin{bmatrix} 5 &7 \\ 10&-2 \end{bmatrix}$

$\begin{bmatrix} 5 &-7 \\ 0&-2 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 5 &-7 \\ 10&-2 \end{bmatrix}$

Explanation:

To solve this problem, subtract the corresponding numbers in each matrix: ( 9-4=5, -2-5=-7, 4-(-6)=10, -5-(-3)=-2) . Then, put those answers in the corresponding places in the answer matrix, so that you get: $\begin{bmatrix} 5 &-7 \\ 10&-2 \end{bmatrix}$ .

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Example Question #6 : Find The Sum Or Difference Of Two Matrices

Add:

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

Possible Answers:

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

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

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

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

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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Precalculus : Sum or Difference of Two Matrices

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Find The Sum Or Difference Of Two Matrices

Example Question #2 : Find The Sum Or Difference Of Two Matrices

Example Question #3 : Find The Sum Or Difference Of Two Matrices

Example Question #3 : Find The Sum Or Difference Of Two Matrices

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

Example Question #1 : Find The Sum Or Difference Of Two Matrices

Example Question #7 : Find The Sum Or Difference Of Two Matrices

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