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Precalculus : Matrices and Vectors

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Example Questions

Example Question #14 : Find The Sum Or Difference Of Two Matrices

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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:

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

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Example Question #21 : 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 A+B .

Possible Answers:

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

DNE

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

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

Correct answer:

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

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.

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

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Example Question #151 : Matrices And Vectors

Let

$A=\begin{pmatrix} 5&1 \\ 3&-7 \end{pmatrix}$ $B=\begin{pmatrix} 1&3 \\ 5&7 \\ 9&11 \end{pmatrix}$

Determine the sum A+B .

Possible Answers:

$\begin{pmatrix} 5&1 \\ 3&-7 \end{pmatrix}$

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

DNE

$\begin{pmatrix} 1&3 \\ 5&7 \\ 9&11 \end{pmatrix}$

Correct answer:

DNE

Explanation:

Because the dimensions of the two matrices are not equal

(A 2x2 matrix is not of the same dimension as a 3x2 matrix)

The sum does not exist.

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Example Question #23 : Find The Sum Or Difference Of Two Matrices

Simplify the following, reducing to one matrix:

$\begin{bmatrix} 6 &-4 &1 \\2 & 6& 5\\-1 &-3 &8 \end{bmatrix}+\begin{bmatrix} 1 & 4 &8 \\ 6& 3& -2\\3 &-2 &7 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 7 &0 &9 \\8 &9 &3 \\2 &-5 &15 \end{bmatrix}$

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

Addition not possible

$\begin{bmatrix} 5 &-8 &-7 \\-4 &3 &7 \\-4 &-1 &1 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 7 &0 &9 \\8 &9 &3 \\2 &-5 &15 \end{bmatrix}$

Explanation:

To add matricies, first check to make sure they are the same size. If they are, then just add corresponding terms in the same spots in each matrix.

Since these are the same size we can add as follows.

$\begin{bmatrix} a &b &c \\d & e& f\\g &h &i \end{bmatrix}+\begin{bmatrix} j & k &l \\ m& n& o\\p &q &r \end{bmatrix}=\begin{bmatrix} a+j &b+k &c+l \\d+m & e+n& f+o\\g+p &h+q &i+r \end{bmatrix}$

$\begin{bmatrix} 6 &-4 &1 \\2 & 6& 5\\-1 &-3 &8 \end{bmatrix}+\begin{bmatrix} 1 & 4 &8 \\ 6& 3& -2\\3 &-2 &7 \end{bmatrix}=\begin{bmatrix} 6+1&-4+4 &1+8 \\2+6& 6+3& 5+-2\\-1+3 &-3+-2 &8+7 \end{bmatrix}$

$\begin{bmatrix} 7 &0 &9 \\8 &9 &3 \\2 &-5 &15 \end{bmatrix}$

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Example Question #24 : Find The Sum Or Difference Of Two Matrices

Find the sum if it exists

$\small \begin{pmatrix} 1&3 \\ 3&2 \end{pmatrix} +\begin{pmatrix} 0&1 \\ 1& 1 \end{pmatrix}$

Possible Answers:

$\small \begin{pmatrix} 1&4 \\ 4& 3 \end{pmatrix}$

$\small \small \begin{pmatrix} 1&2 \\ 2& 1 \end{pmatrix}$

DNE

$\small \small \begin{pmatrix} -1&-2 \\ -2& -1 \end{pmatrix}$

Correct answer:

$\small \begin{pmatrix} 1&4 \\ 4& 3 \end{pmatrix}$

Explanation:

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

$\small \small \begin{pmatrix} 1&3 \\ 3&2 \end{pmatrix} +\begin{pmatrix} 0&1 \\ 1& 1 \end{pmatrix}= \begin{pmatrix} 1+0&3+1 \\ 3+1&2+1 \end{pmatrix}= \begin{pmatrix} 1&4 \\ 4&3 \end{pmatrix}$

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Example Question #25 : Find The Sum Or Difference Of Two Matrices

Sum the two matricies:

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

Possible Answers:

$\begin{bmatrix} 1 &-1 \\0 &7 \end{bmatrix}$

$\begin{bmatrix} 5 &-1 \\-4 &1 \end{bmatrix}$

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

$\begin{bmatrix} 7 &0 \\-1 &1 \end{bmatrix}$

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

Correct answer:

$\begin{bmatrix} 1 &-1 \\0 &7 \end{bmatrix}$

Explanation:

To add matricies, simply add the corresponding terms. Thus,

$\begin{bmatrix} 3 &-1 \\-2 &4 \end{bmatrix} + \begin{bmatrix} -2 &0 \\2 &3 \end{bmatrix}=\begin{bmatrix} 3+(-2)& -1+0\\-2+2 &4+3 \end{bmatrix}=\begin{bmatrix} 1 &-1 \\0 &7 \end{bmatrix}$

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Example Question #26 : Find The Sum Or Difference Of Two Matrices

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

To add two matrices, combine the corresponding numbers in the matrices: ( 1+0=1, 3+5=8, 2+(-2)=0, -1+6=5 ). Then, put those answers in the corresponding spots in the answer matrix. Therefore, your answer is $\begin{bmatrix} 1 &8 \\ 0&5 \end{bmatrix}$ .

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Example Question #152 : Matrices And Vectors

Find the sum of the two matrices

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

Possible Answers:

DNE

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

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

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

Correct answer:

DNE

Explanation:

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

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Example Question #1 : Geometric Vectors

Find the vector given by the product: $\frac{4}{7}<-3,5,14>$

Possible Answers:

<-12,20,56>

$<\frac{-21}{4},\frac{35}{4},\frac{49}{2}>$

$<\frac{-3}{7},\frac{5}{7},2>$

$<\frac{-12}{7},\frac{20}{7},8>$

<-21,35,98>

Correct answer:

$<\frac{-12}{7},\frac{20}{7},8>$

Explanation:

Given a scalar k and a vector v, the vector given by their products is defined component-wise:

k<v_1,v_2,...,v_n>=<kv_1,kv_2,...,kv_n> .

Here, our product is:

$\frac{4}{7}<-3,5,14>=<\frac{4}{7}(-3),\frac{4}{7}(5),\frac{4}{7}(14)>\\ \\ =<\frac{-12}{7},\frac{20}{7},8>$

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Example Question #1 : Geometric Vectors

This question refers to the previous question.

Simplify.

$4\cdot \left<3,6,9\right>$

Possible Answers:

$\left<12, 4, 6\right>$

$\left<1, 24, 3\right>$

$\left<12, 24, 36\right>$

$\left<2, 2, 3\right>$

$\left<12, 24, 6\right>$

Correct answer:

$\left<12, 24, 36\right>$

Explanation:

In order to simplify this problem we need to multiply the scalar factor to each component of the vector.

In our case the scalar factor is

Thus,

$4\cdot \left<3,6,9\right>\rightarrow \left<4\cdot 3, 4\cdot 6, 4\cdot 9\right>$

$\rightarrow \left<12, 24, 36\right>$

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Precalculus : Matrices and Vectors

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #14 : Find The Sum Or Difference Of Two Matrices

Example Question #21 : Find The Sum Or Difference Of Two Matrices

Example Question #151 : Matrices And Vectors

Example Question #23 : Find The Sum Or Difference Of Two Matrices

Example Question #24 : Find The Sum Or Difference Of Two Matrices

Example Question #25 : Find The Sum Or Difference Of Two Matrices

Example Question #26 : Find The Sum Or Difference Of Two Matrices

Example Question #152 : Matrices And Vectors

Example Question #1 : Geometric Vectors

Example Question #1 : Geometric Vectors

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