## 17Calculus Precalculus - Matrix Multiplication

As discussed on the basics of matrices page, multiplying two matrices together is not as simple as multiplying two numbers together. Here is a good introduction video and example. He explains quite thoroughly how to do matrix multiplication.

### Khan Academy - Matrix multiplication introduction [6min-25secs]

Let's work a few practice problems.

Unless otherwise instructed, perform matrix multiplication on these matrices.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right] \left[\begin{array}{rr} 5 & 6 \\ 7 & 8 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right] \left[\begin{array}{rr} 5 & 6 \\ 7 & 8 \end{array}\right]$$

$$\left[\begin{array}{rr} 19 & 22 \\ 43 & 50 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right] \left[\begin{array}{rr} 5 & 6 \\ 7 & 8 \end{array}\right]$$

Solution

### 2015 video

video by PatrickJMT

$$\left[\begin{array}{rr} 19 & 22 \\ 43 & 50 \end{array}\right]$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{ \left[\begin{array}{rr} 2 & 1 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \left[\begin{array}{rrr} 1 & 3 & 6 \\ 2 & 4 & 5 \end{array}\right] }$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{ \left[\begin{array}{rr} 2 & 1 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \left[\begin{array}{rrr} 1 & 3 & 6 \\ 2 & 4 & 5 \end{array}\right] }$$

$$\displaystyle{\left[\begin{array}{rrr} 4 & 10 & 17 \\ 11 & 25 & 38 \\ 17 & 39 & 60\end{array}\right]}$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{ \left[\begin{array}{rr} 2 & 1 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \left[\begin{array}{rrr} 1 & 3 & 6 \\ 2 & 4 & 5 \end{array}\right] }$$

Solution

### 2016 video

video by PatrickJMT

$$\displaystyle{\left[\begin{array}{rrr} 4 & 10 & 17 \\ 11 & 25 & 38 \\ 17 & 39 & 60\end{array}\right]}$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & 5\end{array}\right] \left[\begin{array}{r} 1 \\ 2 \\ -4\end{array}\right] }$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & 5\end{array}\right] \left[\begin{array}{r} 1 \\ 2 \\ -4\end{array}\right] }$$

$$\displaystyle{\left[\begin{array}{r}-7 \\ -22\end{array}\right]}$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & 5\end{array}\right] \left[\begin{array}{r} 1 \\ 2 \\ -4\end{array}\right] }$$

Solution

### 2018 video

video by PatrickJMT

$$\displaystyle{\left[\begin{array}{r}-7 \\ -22\end{array}\right]}$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{\left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] \left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] }$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{\left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] \left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] }$$

$$\displaystyle{\left[\begin{array}{rr} 3 & 13 \\ 51 & 23\end{array}\right]}$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{\left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] \left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] }$$

Solution

### 2019 video

video by MIP4U

$$\displaystyle{\left[\begin{array}{rr} 3 & 13 \\ 51 & 23\end{array}\right]}$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{\left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] \left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] }$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{\left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] \left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] }$$

$$\displaystyle{\left[\begin{array}{rrr} -2 & -17 & 15 \\ -20 & 6 & -26 \\ 22 & 0 & 22\end{array}\right]}$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\displaystyle{\left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] \left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] }$$

Solution

### 2020 video

video by MIP4U

$$\displaystyle{\left[\begin{array}{rrr} -2 & -17 & 15 \\ -20 & 6 & -26 \\ 22 & 0 & 22\end{array}\right]}$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$A = \left[\begin{array}{rrr} 3 & 1 & 4 \end{array}\right]$$
$$B = \left[\begin{array}{rr} 4 & 3 \\ 2 & 5 \\ 6 & 8 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$A = \left[\begin{array}{rrr} 3 & 1 & 4 \end{array}\right]$$
$$B = \left[\begin{array}{rr} 4 & 3 \\ 2 & 5 \\ 6 & 8 \end{array}\right]$$

$$A \cdot B = \left[\begin{array}{rr} 38 & 46 \end{array} \right]$$
$$B \cdot A$$ does not make sense.

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$A = \left[\begin{array}{rrr} 3 & 1 & 4 \end{array}\right]$$
$$B = \left[\begin{array}{rr} 4 & 3 \\ 2 & 5 \\ 6 & 8 \end{array}\right]$$

Solution

### 2877 video

$$A \cdot B = \left[\begin{array}{rr} 38 & 46 \end{array} \right]$$
$$B \cdot A$$ does not make sense.

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, calculate $$A \cdot B$$ using these matrices.
$$A = \left[\begin{array}{rr} 3 & 4 \\ 7 & 2 \\ 5 & 9 \end{array}\right]$$
$$B = \left[\begin{array}{rrr} 3 & 1 & 5 \\ 6 & 9 & 7 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, calculate $$A \cdot B$$ using these matrices.
$$A = \left[\begin{array}{rr} 3 & 4 \\ 7 & 2 \\ 5 & 9 \end{array}\right]$$
$$B = \left[\begin{array}{rrr} 3 & 1 & 5 \\ 6 & 9 & 7 \end{array}\right]$$

$$A \cdot B = \left[\begin{array}{rrr} 33 & 39 & 43 \\ 33 & 25 & 49 \\ 69 & 86 & 88 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, calculate $$A \cdot B$$ using these matrices.
$$A = \left[\begin{array}{rr} 3 & 4 \\ 7 & 2 \\ 5 & 9 \end{array}\right]$$
$$B = \left[\begin{array}{rrr} 3 & 1 & 5 \\ 6 & 9 & 7 \end{array}\right]$$

Solution

### 2878 video

$$A \cdot B = \left[\begin{array}{rrr} 33 & 39 & 43 \\ 33 & 25 & 49 \\ 69 & 86 & 88 \end{array}\right]$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$A = \left[ \begin{array}{rrr} 2 & 5 & 6 \end{array} \right]$$
$$B = \left[\begin{array}{r} 3 \\ 4 \\ -5 \end{array} \right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$A = \left[ \begin{array}{rrr} 2 & 5 & 6 \end{array} \right]$$
$$B = \left[\begin{array}{r} 3 \\ 4 \\ -5 \end{array} \right]$$

$$AB = [ -4 ]$$
$$BA = \left[\begin{array}{rrr} 6 & 15 & 18 \\ 8 & 20 & 24 \\ -10 & -25 & -30 \end{array} \right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$A = \left[ \begin{array}{rrr} 2 & 5 & 6 \end{array} \right]$$
$$B = \left[\begin{array}{r} 3 \\ 4 \\ -5 \end{array} \right]$$

Solution

### 2879 video

$$AB = [ -4 ]$$
$$BA = \left[\begin{array}{rrr} 6 & 15 & 18 \\ 8 & 20 & 24 \\ -10 & -25 & -30 \end{array} \right]$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$A = \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 5 & -6 \end{array}\right]$$
$$B = \left[\begin{array}{rrrr} 5 & 2 & 8 & -1 \\ 3 & 6 & 4 & 5 \\ -2 & 9 & 7 & -3 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$A = \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 5 & -6 \end{array}\right]$$
$$B = \left[\begin{array}{rrrr} 5 & 2 & 8 & -1 \\ 3 & 6 & 4 & 5 \\ -2 & 9 & 7 & -3 \end{array}\right]$$

$$AB = \left[\begin{array}{rrrr} 21 & 8 & 10 & 25 \\ 42 & -18 & 2 & 40 \end{array} \right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$A = \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 5 & -6 \end{array}\right]$$
$$B = \left[\begin{array}{rrrr} 5 & 2 & 8 & -1 \\ 3 & 6 & 4 & 5 \\ -2 & 9 & 7 & -3 \end{array}\right]$$

Solution

### 2880 video

$$AB = \left[\begin{array}{rrrr} 21 & 8 & 10 & 25 \\ 42 & -18 & 2 & 40 \end{array} \right]$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rrr} 2 & 1 & 3 \\ 0 & -1 & 4 \end{array}\right]$$ $$\left[\begin{array}{r} 1 \\ 0 \\ 5 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rrr} 2 & 1 & 3 \\ 0 & -1 & 4 \end{array}\right]$$ $$\left[\begin{array}{r} 1 \\ 0 \\ 5 \end{array}\right]$$

$$\left[\begin{array}{r} 17 \\ 20 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rrr} 2 & 1 & 3 \\ 0 & -1 & 4 \end{array}\right]$$ $$\left[\begin{array}{r} 1 \\ 0 \\ 5 \end{array}\right]$$

Solution

### 2881 video

$$\left[\begin{array}{r} 17 \\ 20 \end{array}\right]$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rr} 1 & 7 \\ -2 & 0 \end{array}\right]$$ $$\left[\begin{array}{rr} 5 & 1 \\ 3 & 2 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rr} 1 & 7 \\ -2 & 0 \end{array}\right]$$ $$\left[\begin{array}{rr} 5 & 1 \\ 3 & 2 \end{array}\right]$$

$$\left[\begin{array}{rr} 26 & 15 \\ -10 & -2 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rr} 1 & 7 \\ -2 & 0 \end{array}\right]$$ $$\left[\begin{array}{rr} 5 & 1 \\ 3 & 2 \end{array}\right]$$

Solution

### 2882 video

$$\left[\begin{array}{rr} 26 & 15 \\ -10 & -2 \end{array}\right]$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rr} 1 & 2 \end{array}\right]$$ $$\left[\begin{array}{rr} 3 & -1 \\ 0 & 2 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rr} 1 & 2 \end{array}\right]$$ $$\left[\begin{array}{rr} 3 & -1 \\ 0 & 2 \end{array}\right]$$

$$\left[\begin{array}{rr} 3 & 3 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rr} 1 & 2 \end{array}\right]$$ $$\left[\begin{array}{rr} 3 & -1 \\ 0 & 2 \end{array}\right]$$

Solution

### 2883 video

$$\left[\begin{array}{rr} 3 & 3 \end{array}\right]$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rrr} 2 & 1 & 0 \\ 3 & 0 & -1 \end{array}\right]$$ $$\left[\begin{array}{rr} 1 & 4 \\ 0 & 2 \\ 5 & 1 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rrr} 2 & 1 & 0 \\ 3 & 0 & -1 \end{array}\right]$$ $$\left[\begin{array}{rr} 1 & 4 \\ 0 & 2 \\ 5 & 1 \end{array}\right]$$

$$\left[\begin{array}{rr} 2 & 10 \\ -2 & 11 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rrr} 2 & 1 & 0 \\ 3 & 0 & -1 \end{array}\right]$$ $$\left[\begin{array}{rr} 1 & 4 \\ 0 & 2 \\ 5 & 1 \end{array}\right]$$

Solution

### 2884 video

$$\left[\begin{array}{rr} 2 & 10 \\ -2 & 11 \end{array}\right]$$

Log in to rate this practice problem and to see it's current rating.

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rrr} -2 & 3 & 4 \\ 4 & 0 & 1 \end{array}\right]$$ $$\left[\begin{array}{rrrr} 0 & -3 & 4 & -1 \\ 1 & 3 & 0 & -5 \\ -4 & 4 & 3 & -2 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rrr} -2 & 3 & 4 \\ 4 & 0 & 1 \end{array}\right]$$ $$\left[\begin{array}{rrrr} 0 & -3 & 4 & -1 \\ 1 & 3 & 0 & -5 \\ -4 & 4 & 3 & -2 \end{array}\right]$$

$$\left[\begin{array}{rrrr} 19 & -1 & -20 & -5 \\ 4 & -16 & 13 & -2 \end{array}\right]$$

Problem Statement

Unless otherwise instructed, perform matrix multiplication on these matrices.
$$\left[\begin{array}{rrr} -2 & 3 & 4 \\ 4 & 0 & 1 \end{array}\right]$$ $$\left[\begin{array}{rrrr} 0 & -3 & 4 & -1 \\ 1 & 3 & 0 & -5 \\ -4 & 4 & 3 & -2 \end{array}\right]$$

Solution

### 2885 video

$$\left[\begin{array}{rrrr} 19 & -1 & -20 & -5 \\ 4 & -16 & 13 & -2 \end{array}\right]$$

Log in to rate this practice problem and to see it's current rating.

### matrix multiplication 17calculus youtube playlist

Really UNDERSTAND Precalculus

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia] Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

free ideas to save on books

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

Math Word Problems Demystified Shop eBags.com, the leading online retailer of luggage, handbags, backpacks, accessories, and more! Shop Amazon - Used Textbooks - Save up to 90% As an Amazon Associate I earn from qualifying purchases.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.