Question 202458
# 1


First, convert the system



{{{system(5x + 3y = 3,2x - 6y = 30)}}}



into the matrix equation



{{{(matrix(2,2,5,3,2,-6))(matrix(2,1,x,y))=(matrix(2,1,3,30))}}}



So what we now need is inverse matrix of {{{(matrix(2,2,5,3,2,-6))}}} to isolate {{{(matrix(2,1,x,y))}}}



*[invoke inverse_of_2x2_matrix 5,3,2,-6]



Now left multiply both sides by the inverse matrix to get



{{{(matrix(2,2,1/6,1/12,1/18,-5/36))(matrix(2,2,5,3,2,-6))(matrix(2,1,x,y))=(matrix(2,2,1/6,1/12,1/18,-5/36))(matrix(2,1,3,30))}}}




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




{{{(matrix(2,1,x,y))=(matrix(2,2,3,-4))}}}



So this means that {{{x=3}}} and {{{y=-4}}}



Note: let me know if you need help with matrix multiplication.




<hr>


# 2


Simply add the corresponding components:



{{{(matrix(2,1,4,7))+(matrix(2,1,8,2))=(matrix(2,1,4+8,7+2))=(matrix(2,1,12,9))}}}



So {{{(matrix(2,1,4,7))+(matrix(2,1,8,2))=(matrix(2,1,12,9))}}}




<hr>


# 3



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


So the final resulting matrix will look like:



{{{(matrix(2,2,x,x,x,x))}}}



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




--------------------------------------------------





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: {{{(matrix(2,2,highlight(1),highlight(8),0,7))(matrix(2,2,highlight(7),6,highlight(7),4))}}}

{{{(1)*(7)+(8)*(7)=7+56=63}}}



 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 {{{63}}}. Now let's update the matrix:

 

{{{(matrix(2,2,63,x,x,x))}}}

--------------------------------------------------





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: {{{(matrix(2,2,highlight(1),highlight(8),0,7))(matrix(2,2,7,highlight(6),7,highlight(4)))}}}

{{{(1)*(6)+(8)*(4)=6+32=38}}}



 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 {{{38}}}. Now let's update the matrix:

 

{{{(matrix(2,2,63,38,x,x))}}}





================================================================================





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: {{{(matrix(2,2,1,8,highlight(0),highlight(7)))(matrix(2,2,highlight(7),6,highlight(7),4))}}}

{{{(0)*(7)+(7)*(7)=0+49=49}}}



 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 {{{49}}}. Now let's update the matrix:

 

{{{(matrix(2,2,63,38,49,x))}}}

--------------------------------------------------





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: {{{(matrix(2,2,1,8,highlight(0),highlight(7)))(matrix(2,2,7,highlight(6),7,highlight(4)))}}}

{{{(0)*(6)+(7)*(4)=0+28=28}}}



 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 {{{28}}}. Now let's update the matrix:

 

{{{(matrix(2,2,63,38,49,28))}}}









==============================================================================



Answer:



So the solution is  {{{(matrix(2,2,63,38,49,28))}}}


In other words,


{{{(matrix(2,2,1,8,0,7))*(matrix(2,2,7,6,7,4))=(matrix(2,2,63,38,49,28))}}}