Multiply Matrices 2

Question 1 of 4

Solve

$[[1,2,0],[0,-1,3],[0,1,0]]times[[3],[4],[1]]$

1. $[[11,-1,4]]$
2. $[[11],[-1],[4]]$
3. $[[11,-7,0]]$
4. $[[11],[-7],[4]]$

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Multiplying Two Matrices

$[[a,b,c]]xx[[p],[q],[r]]=ap+bq+cr$

Two matrices can be multiplied only if the number of columns (

$n$ ) in the first matrix is equal to the number of rows
(

$m$ ) in the second matrix.

First, check the dimensions of each matrix

Matrix

$1$ :

$[[1,2,0],[0,-1,3],[0,1,0]]$

rows(

$m$ )

$=3$

columns(

$n$ )

$=3$

Dimension

$(mtimesn)=3xx$ $3$

Matrix

$2$ :

$[[3],[4],[1]]$

rows(

$m$ )

$=3$

columns(

$n$ )

$=1$

Dimension

$(mtimesn)=$ $3$

$xx1$

Since the number of columns in the first matrix ( $3$ ) and the number of rows in the second matrix
( $3$ ) are equal, these two matrices can be multiplied

Next, proceed with multiplying the two matrices

		$[[1,2,0],[0,-1,3],[0,1,0]]times$ $[[3],[4],[1]]$

	$=$	$\begin{bmatrix} (1\cdot\color{#9a00c7}{3})+(2\cdot\color{#9a00c7}{4})+(0\cdot\color{#9a00c7}{1}) \\[0.3em] (0\cdot\color{#9a00c7}{3})+(-1\cdot\color{#9a00c7}{4})+(3\cdot\color{#9a00c7}{1}) \\[0.3em] (0\cdot\color{#9a00c7}{3})+(1\cdot\color{#9a00c7}{4})+(0\cdot\color{#9a00c7}{1}) \end{bmatrix}$

	$=$	$[[3+8+0],[0+(-4)+3],[0+4+0]]$

	$=$	$[[11],[-1],[4]]$

$[[11],[-1],[4]]$

Question 2 of 4

Solve

$[[4,2,1],[3,-5,2]]times[[3,2,4],[1,4,5],[2,6,1]]$

1. $[[14,8],[2,22],[-27,11]]$
2. $[[16,22,20],[-8,2,11]]$
3. $[[16,22,27],[8,-2,-11]]$
4. $[[16,8],[22,-2],[27,-11]]$

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Multiplying Two Matrices

$[[a,b,c]]xx[[p],[q],[r]]=ap+bq+cr$

Two matrices can be multiplied only if the number of columns (

$n$ ) in the first matrix is equal to the number of rows
(

$m$ ) in the second matrix.

First, check the dimensions of each matrix

Matrix

$1$ :

$[[4,2,1],[3,-5,2]]$

rows(

$m$ )

$=2$

columns(

$n$ )

$=3$

Dimension

$(mtimesn)=2xx$ $3$

Matrix

$2$ :

$[[3,2,4],[1,4,5],[2,6,1]]$

rows(

$m$ )

$=3$

columns(

$n$ )

$=3$

Dimension

$(mtimesn)=$ $3$

$xx3$

Since the number of columns in the first matrix ( $3$ ) and the number of rows in the second matrix
( $3$ ) are equal, these two matrices can be multiplied

Next, proceed with multiplying the two matrices

		$[[4,2,1],[3,-5,2]]times$ $[[3,2,4],[1,4,5],[2,6,1]]$

	$=$	$\begin{bmatrix} (4\cdot\color{#9a00c7}{3})+(2\cdot\color{#9a00c7}{1})+(1\cdot\color{#9a00c7}{2}) & (4\cdot\color{#9a00c7}{2})+(2\cdot\color{#9a00c7}{4})+(1\cdot\color{#9a00c7}{6}) & (4\cdot\color{#9a00c7}{4})+(2\cdot\color{#9a00c7}{5})+(1\cdot\color{#9a00c7}{1}) \\[0.3em] (3\cdot\color{#9a00c7}{3})+(-5\cdot\color{#9a00c7}{1})+(2\cdot\color{#9a00c7}{2}) & (3\cdot\color{#9a00c7}{2})+(-5\cdot\color{#9a00c7}{4})+(2\cdot\color{#9a00c7}{6}) & (3\cdot\color{#9a00c7}{4})+(-5\cdot\color{#9a00c7}{5})+(2\cdot\color{#9a00c7}{1}) \end{bmatrix}$

	$=$	$[[12+2+2,8+8+6,16+10+1],[9+(-5)+4,6+(-20)+12,12+(-25)+2]]$

	$=$	$[[16,22,27],[8,-2,-11]]$

$[[16,22,27],[8,-2,-11]]$

Question 3 of 4

Solve for

$2xy$ , given that:

$x=[[3,-1],[5,-2]]$

$y=[[1,0],[-4,3]]$

1. $[[14,-6],[26,-12]]$
2. $[[14,-6,26,-12]]$
3. $[[14,-26],[-6,-12]]$
4. $[[14],[6],[26],[12]]$

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Multiplying Two Matrices

$[[a,b,c]]xx[[p],[q],[r]]=ap+bq+cr$

Two matrices can be multiplied only if the number of columns (

$n$ ) in the first matrix is equal to the number of rows
(

$m$ ) in the second matrix.

First, check the dimensions of matrices

$x$ and

$y$

Matrix

$x$ :

$[[3,-1],[5,-2]]$

rows(

$m$ )

$=2$

columns(

$n$ )

$=2$

Dimension

$(mtimesn)=2xx$ $2$

Matrix

$y$ :

$[[1,0],[-4,3]]$

rows(

$m$ )

$=2$

columns(

$n$ )

$=2$

Dimension

$(mtimesn)=$ $2$

$xx2$

Since the number of columns in the first matrix ( $2$ ) and the number of rows in the second matrix
( $2$ ) are equal, these two matrices can be multiplied

Next, substitute the two matrices to

$2xy$ and solve

$2xy$	$=$	$2times[[3,-1],[5,-2]]times$ $[[1,0],[-4,3]]$

	$=$	$[[6,-2],[10,-4]]times$ $[[1,0],[-4,3]]$

	$=$	$\begin{bmatrix} (6\cdot\color{#9a00c7}{1})+(-2\cdot\color{#9a00c7}{-4}) & (6\cdot\color{#9a00c7}{0})+(-2\cdot\color{#9a00c7}{3}) \\[0.3em] (10\cdot\color{#9a00c7}{1})+(-4\cdot\color{#9a00c7}{-4}) & (10\cdot\color{#9a00c7}{0})+(-4\cdot\color{#9a00c7}{3}) \end{bmatrix}$

	$=$	$[[6+8,0+(-6)],[10+16,0+(-12)]]$

	$=$	$[[14,-6],[26,-12]]$

$[[14,-6],[26,-12]]$

Question 4 of 4

Solve for

$x(x+y)$ , given that:

$x=[[3,-1],[5,-2]]$

$y=[[1,0],[-4,3]]$

1. $[[11,18],[-4,-7]]$
2. $[[11],[4],[18],[7]]$
3. $[[11,-4],[18,-7]]$
4. $[[-11,4,18,-7]]$

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Multiplying Two Matrices

$[[a,b,c]]xx[[p],[q],[r]]=ap+bq+cr$

Two matrices can be multiplied only if the number of columns (

$n$ ) in the first matrix is equal to the number of rows
(

$m$ ) in the second matrix.

Two matrices can be added only if their dimensions (

$mtimesn$ ) are equal.

First, check the dimensions of matrices

$x$ and

$y$

Matrix

$x$ :

$[[3,-1],[5,-2]]$

rows(

$m$ )

$=2$

columns(

$n$ )

$=2$

Dimension

$(mtimesn)=2xx$ $2$

Matrix

$y$ :

$[[1,0],[-4,3]]$

rows(

$m$ )

$=2$

columns(

$n$ )

$=2$

Dimension

$(mtimesn)=$ $2$

$xx2$

Since the number of columns in the first matrix ( $2$ ) and the number of rows in the second matrix
( $2$ ) are equal, these two matrices can be multiplied

Also note that since their dimensions are equal, they can be added

Next, substitute the two matrices to

$x(x+y)$ and solve

$x(x+y)$	$=$	$[[3,-1],[5,-2]]times([[3,-1],[5,-2]]+[[1,0],[-4,3]])$

	$=$	$[[3,-1],[5,-2]]times[[3+1,-1+0],[5+(-4),-2+3]]$

	$=$	$[[3,-1],[5,-2]]times$ $[[4,-1],[1,1]]$

	$=$	$\begin{bmatrix} (3\cdot\color{#9a00c7}{4})+(-1\cdot\color{#9a00c7}{1}) & (3\cdot\color{#9a00c7}{-1})+(-1\cdot\color{#9a00c7}{1}) \\[0.3em] (5\cdot\color{#9a00c7}{4})+(-2\cdot\color{#9a00c7}{1}) & (5\cdot\color{#9a00c7}{-1})+(-2\cdot\color{#9a00c7}{1}) \end{bmatrix}$

	$=$	$[[12+(-1),-3+(-1)],[20+(-2),-5+(-2)]]$

	$=$	$[[11,-4],[18,-7]]$

$[[11,-4],[18,-7]]$

Multiply Matrices 2

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Quiz summary

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1. Question

Hint

Keep Going!

Multiplying Two Matrices

2. Question

Hint

Correct!

Multiplying Two Matrices

3. Question

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Great Work!

Multiplying Two Matrices

4. Question

Hint

Excellent!

Multiplying Two Matrices

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