Question 200660
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Table of Contents:

<a href="#1">Part i)</a>
<a href="#2">Part ii)</a> 
<a href="#3">Part iii)</a>
<a href="#4">Part iv)</a>



Given Matrices:



{{{A=(matrix(2,3,2,3,0,1,4,-3))}}}



{{{B=(matrix(2,3,1,-1,4,2,3,5))}}}



{{{C=(matrix(2,2,2,-1,3,5))}}}




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Part i)



{{{A+B=(matrix(2,3,2,3,0,1,4,-3))+(matrix(2,3,1,-1,4,2,3,5))}}} Start with the addition of the two given matrices



{{{A+B=(matrix(2,3,2+1,3+(-1),0+4,1+2,4+3,-3+5))}}} Add up the matrices by adding the corresponding entries



{{{A+B=(matrix(2,3,3,2,4,3,7,2))}}} Add


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Answer:


So {{{A+B=(matrix(2,3,3,2,4,3,7,2))}}}


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Part ii)


To find the inverse of the matrix {{{C=(matrix(2,2,2,-1,3,5))}}}, we can follow these steps:



Step 1) Find the determinant



The <a href="http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver">determinant</a> of {{{(matrix(2,2,2,-1,3,5))}}} is {{{abs(matrix(2,2,2,-1,3,5))=13}}}. So this means that {{{d=13}}}


note: since the determinant is NOT equal to zero, the matrix inverse exists.


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Step 2) Swap the values



Now switch the highlighted values {{{(matrix(2,2,highlight(2),-1,3,highlight(5)))}}} to get {{{(matrix(2,2,highlight(5),-1,3,highlight(2)))}}}



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Step 3) Change the sign



Now change the sign of the highlighted values {{{(matrix(2,2,5,highlight(-1),highlight(3),2))}}} to get {{{(matrix(2,2,5,highlight(1),highlight(-3),2))}}}



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Step 4) Multiply by the inverse of the determinant



Multiply by {{{1/d}}} to get {{{(1/d)(matrix(2,2,5,1,-3,2))}}}



Plug in {{{d=13}}} to get {{{(1/13)(matrix(2,2,5,1,-3,2))}}}



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Step 5) Perform scalar multiplication and simplify.



Multiply {{{1/13}}} by EVERY element to get {{{(matrix(2,2,(1/13)(5),(1/13)(1),(1/13)(-3),(1/13)(2)))}}}



Multiply to get {{{(matrix(2,2,5/13,1/13,-3/13,2/13))}}}



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Answer:

So the inverse of {{{(matrix(2,2,2,-1,3,5))}}} is {{{(matrix(2,2,5/13,1/13,-3/13,2/13))}}}



This means that if {{{C=(matrix(2,2,2,-1,3,5))}}} then {{{C^(-1)=(matrix(2,2,5/13,1/13,-3/13,2/13))}}}



note: to verify your work, multiply the matrices {{{C}}} and {{{C^(-1)}}} and you should get the matrix {{{(matrix(2,2,1,0,0,1))}}} (the identity matrix).


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Part iii)


Take note that the dimensions of matrices A and B are 2x3 and 2x3 respectively. Since the inner dimensions do NOT match, this means that the matrix product AB is NOT defined. In other words, AB does NOT exist.


note: if A and B are mxn and nxp matrices, then the product AB is defined. It is only defined if the number of columns of A equals the number of rows of B. Otherwise, AB does NOT exist.


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Answer:


So the matrix product AB is not defined (ie it doesn't exist).




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Part iv)


I'm sure you meant to write {{{B^T*C}}} correct? If so, then...



Step 1: Transpose matrix B to get {{{B^T}}}. You do this by converting each row into a column (and vice versa).


{{{B^T = (matrix(2,3,1,-1,4,2,3,5))^T = (matrix(3,2,1,2-1,3,4,5)) }}}



Step 2: Multiply matrices {{{B^T}}} and C. Since the inner dimensions are equal, this means that the matrix product {{{B^T*C}}} is defined.




Since the first matrix is a 3 by 2 matrix and the second matrix is a 2 by 2 matrix, this means that the resulting matrix will be a 3 by 2 matrix.


So the final resulting matrix will look like:



{{{B^T*C=(matrix(3,2,x,x,x,x,x,x))}}}



note: the "x"s are just placeholders for now




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Multiply the corresponding entries from the <font size="4" color="red">1st</font> row of the first matrix by the <font size="4" color="green">1st</font> column of the second matrix. After multiplying, add the values:



<font size="4" color="red">1st</font> row, <font size="4" color="green">1st</font> column: {{{B^T*C=(matrix(3,2,highlight(1),highlight(2),-1,3,4,5))(matrix(2,2,highlight(2),-1,highlight(3),5))}}}

{{{(1)*(2)+(2)*(3)=2+6=8}}}



 So the element in the <font size="4" color="red">1st</font> row, <font size="4" color="green">1st</font> column of the resulting matrix is {{{8}}}. Now let's update the matrix:

 

{{{B^T*C=(matrix(3,2,8,x,x,x,x,x))}}}

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Multiply the corresponding entries from the <font size="4" color="red">1st</font> row of the first matrix by the <font size="4" color="green">2nd</font> column of the second matrix. After multiplying, add the values:



<font size="4" color="red">1st</font> row, <font size="4" color="green">2nd</font> column: {{{B^T*C=(matrix(3,2,highlight(1),highlight(2),-1,3,4,5))(matrix(2,2,2,highlight(-1),3,highlight(5)))}}}

{{{(1)*(-1)+(2)*(5)=-1+10=9}}}



 So the element in the <font size="4" color="red">1st</font> row, <font size="4" color="green">2nd</font> column of the resulting matrix is {{{9}}}. Now let's update the matrix:

 

{{{B^T*C=(matrix(3,2,8,9,x,x,x,x))}}}





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Multiply the corresponding entries from the <font size="4" color="red">2nd</font> row of the first matrix by the <font size="4" color="green">1st</font> column of the second matrix. After multiplying, add the values:



<font size="4" color="red">2nd</font> row, <font size="4" color="green">1st</font> column: {{{B^T*C=(matrix(3,2,1,2,highlight(-1),highlight(3),4,5))(matrix(2,2,highlight(2),-1,highlight(3),5))}}}

{{{(-1)*(2)+(3)*(3)=-2+9=7}}}



 So the element in the <font size="4" color="red">2nd</font> row, <font size="4" color="green">1st</font> column of the resulting matrix is {{{7}}}. Now let's update the matrix:

 

{{{B^T*C=(matrix(3,2,8,9,7,x,x,x))}}}

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Multiply the corresponding entries from the <font size="4" color="red">2nd</font> row of the first matrix by the <font size="4" color="green">2nd</font> column of the second matrix. After multiplying, add the values:



<font size="4" color="red">2nd</font> row, <font size="4" color="green">2nd</font> column: {{{B^T*C=(matrix(3,2,1,2,highlight(-1),highlight(3),4,5))(matrix(2,2,2,highlight(-1),3,highlight(5)))}}}

{{{(-1)*(-1)+(3)*(5)=1+15=16}}}



 So the element in the <font size="4" color="red">2nd</font> row, <font size="4" color="green">2nd</font> column of the resulting matrix is {{{16}}}. Now let's update the matrix:

 

{{{B^T*C=(matrix(3,2,8,9,7,16,x,x))}}}





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Multiply the corresponding entries from the <font size="4" color="red">3rd</font> row of the first matrix by the <font size="4" color="green">1st</font> column of the second matrix. After multiplying, add the values:



<font size="4" color="red">3rd</font> row, <font size="4" color="green">1st</font> column: {{{B^T*C=(matrix(3,2,1,2,-1,3,highlight(4),highlight(5)))(matrix(2,2,highlight(2),-1,highlight(3),5))}}}


{{{(4)*(2)+(5)*(3)=8+15=23}}}



 So the element in the <font size="4" color="red">3rd</font> row, <font size="4" color="green">1st</font> column of the resulting matrix is {{{23}}}. Now let's update the matrix:

 

{{{B^T*C=(matrix(3,2,8,9,7,16,23,x))}}}

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Multiply the corresponding entries from the <font size="4" color="red">3rd</font> row of the first matrix by the <font size="4" color="green">2nd</font> column of the second matrix. After multiplying, add the values:



<font size="4" color="red">3rd</font> row, <font size="4" color="green">2nd</font> column: {{{B^T*C=(matrix(3,2,1,2,-1,3,highlight(4),highlight(5)))(matrix(2,2,2,highlight(-1),3,highlight(5)))}}}


{{{(4)*(-1)+(5)*(5)=-4+25=21}}}



 So the element in the <font size="4" color="red">3rd</font> row, <font size="4" color="green">2nd</font> column of the resulting matrix is {{{21}}}. Now let's update the matrix:

 

{{{B^T*C = (matrix(3,2,8,9,7,16,23,21))}}}









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Answer:



So {{{B^T*C = (matrix(3,2,1,2,-1,3,4,5))*(matrix(3,2,2,-1,3,5)) = (matrix(3,2,8,9,7,16,23,21))}}}




In other words,


{{{B^T*C = (matrix(3,2,8,9,7,16,23,21))}}}



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