Math / Algebra

QuestionIf $M=\left[\begin{array}{lll}1 & 1 & 3\end{array}\right]$ and $N=\left[\begin{array}{c}-5 \\ 6 \\ 3\end{array}\right]$ , what are $M N$ and $N M$ ? If a matrix exists, select its size before entering your answer. If a matrix does not exist, select undefined. $M N=$ $\square$ $N M=$ $\square$

Studdy Solution

STEP 1

What is this asking? We're multiplying matrices $M$ and $N$ , but in two different orders, to see what we get! Watch out! Matrix multiplication isn't always possible, and even when it is, the order matters!
A $1 \times 3$ matrix times a $3 \times 1$ matrix won't give the same result as a $3 \times 1$ matrix times a $1 \times 3$ matrix.

STEP 2

1. Calculate $MN$
2. Calculate $NM$

STEP 3

$M$ has dimensions $1 \times 3$ and $N$ has dimensions $3 \times 1$ .
Since the number of columns in $M$ ( $3$ ) matches the number of rows in $N$ ( $3$ ), we can multiply them!
The resulting matrix will be $1 \times 1$ .

STEP 4

$\begin{aligned} MN &= \begin{bmatrix} 1 & 1 & 3 \end{bmatrix} \begin{bmatrix} -5 \\ 6 \\ 3 \end{bmatrix} \\ &= -5 + 6 + 9 \\ &= 10 \end{aligned}$ So, $MN$ is a $1 \times 1$ matrix with the single entry $10$ !

STEP 5

$N$ has dimensions $3 \times 1$ and $M$ has dimensions $1 \times 3$ .
Since the number of columns in $N$ ( $1$ ) matches the number of rows in $M$ ( $1$ ), we can multiply these too!
The resulting matrix will have dimensions $3 \times 3$ .

STEP 6

$NM = \begin{bmatrix} -5 \\ 6 \\ 3 \end{bmatrix} \begin{bmatrix} 1 & 1 & 3 \end{bmatrix} = \begin{bmatrix} -5 & -5 & -15 \\ 6 & 6 & 18 \\ 3 & 3 & 9 \end{bmatrix}$ So, $NM$ is a $3 \times 3$ matrix!

STEP 7

$MN = \begin{bmatrix} 10 \end{bmatrix}$
$NM = \begin{bmatrix} -5 & -5 & -15 \\ 6 & 6 & 18 \\ 3 & 3 & 9 \end{bmatrix}$

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Studdy Solution

STEP 1

STEP 2

1. Calculate $MN$
2. Calculate $NM$

STEP 3

$M$ has dimensions $1 \times 3$ and $N$ has dimensions $3 \times 1$ .
Since the number of columns in $M$ ( $3$ ) matches the number of rows in $N$ ( $3$ ), we can multiply them!
The resulting matrix will be $1 \times 1$ .

STEP 4

$\begin{aligned} MN &= \begin{bmatrix} 1 & 1 & 3 \end{bmatrix} \begin{bmatrix} -5 \\ 6 \\ 3 \end{bmatrix} \\ &= -5 + 6 + 9 \\ &= 10 \end{aligned}$ So, $MN$ is a $1 \times 1$ matrix with the single entry $10$ !

STEP 5

$N$ has dimensions $3 \times 1$ and $M$ has dimensions $1 \times 3$ .
Since the number of columns in $N$ ( $1$ ) matches the number of rows in $M$ ( $1$ ), we can multiply these too!
The resulting matrix will have dimensions $3 \times 3$ .

STEP 6

$NM = \begin{bmatrix} -5 \\ 6 \\ 3 \end{bmatrix} \begin{bmatrix} 1 & 1 & 3 \end{bmatrix} = \begin{bmatrix} -5 & -5 & -15 \\ 6 & 6 & 18 \\ 3 & 3 & 9 \end{bmatrix}$ So, $NM$ is a $3 \times 3$ matrix!

STEP 7

$MN = \begin{bmatrix} 10 \end{bmatrix}$
$NM = \begin{bmatrix} -5 & -5 & -15 \\ 6 & 6 & 18 \\ 3 & 3 & 9 \end{bmatrix}$

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Studdy Solution

STEP 1

STEP 2

1. Calculate MNMNMN2. Calculate NMNMNM

STEP 3

MMM has dimensions 1×31 \times 31×3 and NNN has dimensions 3×13 \times 13×1.Since the number of columns in MMM (333) matches the number of rows in NNN (333), we can multiply them!The resulting matrix will be 1×11 \times 11×1.

STEP 4

STEP 5

NNN has dimensions 3×13 \times 13×1 and MMM has dimensions 1×31 \times 31×3.Since the number of columns in NNN (111) matches the number of rows in MMM (111), we can multiply these too!The resulting matrix will have dimensions 3×33 \times 33×3.

STEP 6

STEP 7

MN=[10]MN = \begin{bmatrix} 10 \end{bmatrix}MN=[10​]NM=[−5−5−156618339]NM = \begin{bmatrix} -5 & -5 & -15 \\ 6 & 6 & 18 \\ 3 & 3 & 9 \end{bmatrix}NM=⎣⎡​−563​−563​−15189​⎦⎤​

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1. Calculate $MN$
2. Calculate $NM$

$M$ has dimensions $1 \times 3$ and $N$ has dimensions $3 \times 1$ .
Since the number of columns in $M$ ( $3$ ) matches the number of rows in $N$ ( $3$ ), we can multiply them!
The resulting matrix will be $1 \times 1$ .

$N$ has dimensions $3 \times 1$ and $M$ has dimensions $1 \times 3$ .
Since the number of columns in $N$ ( $1$ ) matches the number of rows in $M$ ( $1$ ), we can multiply these too!
The resulting matrix will have dimensions $3 \times 3$ .

$MN = \begin{bmatrix} 10 \end{bmatrix}$
$NM = \begin{bmatrix} -5 & -5 & -15 \\ 6 & 6 & 18 \\ 3 & 3 & 9 \end{bmatrix}$