## 17Calculus Precalculus - Matrix Multiplication

##### 17Calculus

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.

Practice

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

$$\left[\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right] \left[\begin{array}{rr} 5 & 6 \\ 7 & 8 \end{array}\right]$$

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

Problem Statement

$$\left[\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right] \left[\begin{array}{rr} 5 & 6 \\ 7 & 8 \end{array}\right]$$

Solution

### PatrickJMT - 2015 video solution

video by PatrickJMT

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

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$$\displaystyle{ \left[\begin{array}{rr} 2 & 1 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \left[\begin{array}{rrr} 1 & 3 & 6 \\ 2 & 4 & 5 \end{array}\right] }$$

Problem Statement

$$\displaystyle{ \left[\begin{array}{rr} 2 & 1 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \left[\begin{array}{rrr} 1 & 3 & 6 \\ 2 & 4 & 5 \end{array}\right] }$$

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

Problem Statement

$$\displaystyle{ \left[\begin{array}{rr} 2 & 1 \\ 3 & 4 \\ 5 & 6 \end{array}\right] \left[\begin{array}{rrr} 1 & 3 & 6 \\ 2 & 4 & 5 \end{array}\right] }$$

Solution

### PatrickJMT - 2016 video solution

video by PatrickJMT

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

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$$\displaystyle{ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & 5\end{array}\right] \left[\begin{array}{r} 1 \\ 2 \\ -4\end{array}\right] }$$

Problem Statement

$$\displaystyle{ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & 5\end{array}\right] \left[\begin{array}{r} 1 \\ 2 \\ -4\end{array}\right] }$$

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

Problem Statement

$$\displaystyle{ \left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & 5\end{array}\right] \left[\begin{array}{r} 1 \\ 2 \\ -4\end{array}\right] }$$

Solution

### PatrickJMT - 2018 video solution

video by PatrickJMT

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

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$$\displaystyle{\left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] \left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] }$$

Problem Statement

$$\displaystyle{\left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] \left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] }$$

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

Problem Statement

$$\displaystyle{\left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] \left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] }$$

Solution

### MIP4U - 2019 video solution

video by MIP4U

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

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$$\displaystyle{\left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] \left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] }$$

Problem Statement

$$\displaystyle{\left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] \left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] }$$

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

Problem Statement

$$\displaystyle{\left[\begin{array}{rr} -2 & 3 \\ -4 & -2 \\ 5 & 1 \end{array}\right] \left[\begin{array}{rrr} 4 & 1 & 3 \\ 2 & -5 & 7\end{array}\right] }$$

Solution

### MIP4U - 2020 video solution

video by MIP4U

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

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Calculate AB and BA where
$$A = \left[\begin{array}{rrr} 3 & 1 & 4 \end{array}\right]$$
$$B = \left[\begin{array}{rr} 4 & 3 \\ 2 & 5 \\ 6 & 8 \end{array}\right]$$

Problem Statement

Calculate AB and BA where
$$A = \left[\begin{array}{rrr} 3 & 1 & 4 \end{array}\right]$$
$$B = \left[\begin{array}{rr} 4 & 3 \\ 2 & 5 \\ 6 & 8 \end{array}\right]$$

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

Problem Statement

Calculate AB and BA where
$$A = \left[\begin{array}{rrr} 3 & 1 & 4 \end{array}\right]$$
$$B = \left[\begin{array}{rr} 4 & 3 \\ 2 & 5 \\ 6 & 8 \end{array}\right]$$

Solution

### 2877 video solution

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

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Calculate $$A \cdot B$$ where
$$A = \left[\begin{array}{rr} 3 & 4 \\ 7 & 2 \\ 5 & 9 \end{array}\right]$$
$$B = \left[\begin{array}{rrr} 3 & 1 & 5 \\ 6 & 9 & 7 \end{array}\right]$$

Problem Statement

Calculate $$A \cdot B$$ where
$$A = \left[\begin{array}{rr} 3 & 4 \\ 7 & 2 \\ 5 & 9 \end{array}\right]$$
$$B = \left[\begin{array}{rrr} 3 & 1 & 5 \\ 6 & 9 & 7 \end{array}\right]$$

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

Problem Statement

Calculate $$A \cdot B$$ where
$$A = \left[\begin{array}{rr} 3 & 4 \\ 7 & 2 \\ 5 & 9 \end{array}\right]$$
$$B = \left[\begin{array}{rrr} 3 & 1 & 5 \\ 6 & 9 & 7 \end{array}\right]$$

Solution

### 2878 video solution

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

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Calculate AB and BA where
$$A = \left[ \begin{array}{rrr} 2 & 5 & 6 \end{array} \right]$$
$$B = \left[\begin{array}{r} 3 \\ 4 \\ -5 \end{array} \right]$$

Problem Statement

Calculate AB and BA where
$$A = \left[ \begin{array}{rrr} 2 & 5 & 6 \end{array} \right]$$
$$B = \left[\begin{array}{r} 3 \\ 4 \\ -5 \end{array} \right]$$

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

Problem Statement

Calculate AB and BA where
$$A = \left[ \begin{array}{rrr} 2 & 5 & 6 \end{array} \right]$$
$$B = \left[\begin{array}{r} 3 \\ 4 \\ -5 \end{array} \right]$$

Solution

### 2879 video solution

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

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Calculate AB where
$$A = \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 5 & -6 \end{array}\right]$$
$$B = \left[\begin{array}{rrrr} 5 & 2 & 8 & -1 \\ 3 & 6 & 4 & 5 \\ -2 & 9 & 7 & -3 \end{array}\right]$$

Problem Statement

Calculate AB where
$$A = \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 5 & -6 \end{array}\right]$$
$$B = \left[\begin{array}{rrrr} 5 & 2 & 8 & -1 \\ 3 & 6 & 4 & 5 \\ -2 & 9 & 7 & -3 \end{array}\right]$$

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

Problem Statement

Calculate AB where
$$A = \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 5 & -6 \end{array}\right]$$
$$B = \left[\begin{array}{rrrr} 5 & 2 & 8 & -1 \\ 3 & 6 & 4 & 5 \\ -2 & 9 & 7 & -3 \end{array}\right]$$

Solution

### 2880 video solution

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

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$$\left[\begin{array}{rrr} 2 & 1 & 3 \\ 0 & -1 & 4 \end{array}\right]$$ $$\left[\begin{array}{r} 1 \\ 0 \\ 5 \end{array}\right]$$

Problem Statement

$$\left[\begin{array}{rrr} 2 & 1 & 3 \\ 0 & -1 & 4 \end{array}\right]$$ $$\left[\begin{array}{r} 1 \\ 0 \\ 5 \end{array}\right]$$

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

Problem Statement

$$\left[\begin{array}{rrr} 2 & 1 & 3 \\ 0 & -1 & 4 \end{array}\right]$$ $$\left[\begin{array}{r} 1 \\ 0 \\ 5 \end{array}\right]$$

Solution

### 2881 video solution

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

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$$\left[\begin{array}{rr} 1 & 7 \\ -2 & 0 \end{array}\right]$$ $$\left[\begin{array}{rr} 5 & 1 \\ 3 & 2 \end{array}\right]$$

Problem Statement

$$\left[\begin{array}{rr} 1 & 7 \\ -2 & 0 \end{array}\right]$$ $$\left[\begin{array}{rr} 5 & 1 \\ 3 & 2 \end{array}\right]$$

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

Problem Statement

$$\left[\begin{array}{rr} 1 & 7 \\ -2 & 0 \end{array}\right]$$ $$\left[\begin{array}{rr} 5 & 1 \\ 3 & 2 \end{array}\right]$$

Solution

### 2882 video solution

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

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$$\left[\begin{array}{rr} 1 & 2 \end{array}\right]$$ $$\left[\begin{array}{rr} 3 & -1 \\ 0 & 2 \end{array}\right]$$

Problem Statement

$$\left[\begin{array}{rr} 1 & 2 \end{array}\right]$$ $$\left[\begin{array}{rr} 3 & -1 \\ 0 & 2 \end{array}\right]$$

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

Problem Statement

$$\left[\begin{array}{rr} 1 & 2 \end{array}\right]$$ $$\left[\begin{array}{rr} 3 & -1 \\ 0 & 2 \end{array}\right]$$

Solution

### 2883 video solution

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

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$$\left[\begin{array}{rrr} 2 & 1 & 0 \\ 3 & 0 & -1 \end{array}\right]$$ $$\left[\begin{array}{rr} 1 & 4 \\ 0 & 2 \\ 5 & 1 \end{array}\right]$$

Problem Statement

$$\left[\begin{array}{rrr} 2 & 1 & 0 \\ 3 & 0 & -1 \end{array}\right]$$ $$\left[\begin{array}{rr} 1 & 4 \\ 0 & 2 \\ 5 & 1 \end{array}\right]$$

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

Problem Statement

$$\left[\begin{array}{rrr} 2 & 1 & 0 \\ 3 & 0 & -1 \end{array}\right]$$ $$\left[\begin{array}{rr} 1 & 4 \\ 0 & 2 \\ 5 & 1 \end{array}\right]$$

Solution

### 2884 video solution

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

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$$\left[\begin{array}{rrr} -2 & 3 & 4 \\ 4 & 0 & 1 \end{array}\right]$$ $$\left[\begin{array}{rrrr} 0 & -3 & 4 & -1 \\ 1 & 3 & 0 & -5 \\ -4 & 4 & 3 & -2 \end{array}\right]$$

Problem Statement

$$\left[\begin{array}{rrr} -2 & 3 & 4 \\ 4 & 0 & 1 \end{array}\right]$$ $$\left[\begin{array}{rrrr} 0 & -3 & 4 & -1 \\ 1 & 3 & 0 & -5 \\ -4 & 4 & 3 & -2 \end{array}\right]$$

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

Problem Statement

$$\left[\begin{array}{rrr} -2 & 3 & 4 \\ 4 & 0 & 1 \end{array}\right]$$ $$\left[\begin{array}{rrrr} 0 & -3 & 4 & -1 \\ 1 & 3 & 0 & -5 \\ -4 & 4 & 3 & -2 \end{array}\right]$$

Solution

### 2885 video solution

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

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