SOLUTION: Can you help me solve the matrix equation. {{{matrix(1,3,(matrix(2,2,2,4,0,1))X, "=", (matrix(2,3,4,0,6,3,-1,5)))}}}

Algebra ->  Matrices-and-determiminant -> SOLUTION: Can you help me solve the matrix equation. {{{matrix(1,3,(matrix(2,2,2,4,0,1))X, "=", (matrix(2,3,4,0,6,3,-1,5)))}}}      Log On


   

Question 176018This question is from textbook Algebra 2
: Can you help me solve the matrix equation.
 This question is from textbook Algebra 2

Found 3 solutions by Mathtut, jim_thompson5910, Edwin McCravy:
Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29%28matrix%282%2C3%2C4%2C0%2C6%2C3%2C-1%2C5%29%29= %28matrix%282%2C3%2C-4%2C4%2C-8%2C-3%2C1%2C-5%29%29
:


row 1 and column 1 signified by r1cl
:
r1c1=2(4)-4(3)=-4
r1c2=2(0)-4(-1)=4
r1c3=2(6)-4(5)=-8
r2c1=0(4)-1(3)=-3
r2c2=0(0)-1(-1)=1
r2c3=0(6)-1(5)=-5
:


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, we need to find the inverse of the matrix %28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29


Solved by pluggable solver: Finding the Inverse of a 2x2 Matrix

To find the inverse of the matrix A=%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29, we can follow these steps:

Step 1) Find the determinant



The determinant of %28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29 is abs%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29=2. So this means that d=2

Step 2) Swap the values



Now switch the highlighted values %28matrix%282%2C2%2Chighlight%282%29%2C4%2C0%2Chighlight%281%29%29%29 to get %28matrix%282%2C2%2Chighlight%281%29%2C4%2C0%2Chighlight%282%29%29%29

Step 3) Change the sign



Now change the sign of the highlighted values %28matrix%282%2C2%2C1%2Chighlight%284%29%2Chighlight%280%29%2C2%29%29 to get %28matrix%282%2C2%2C1%2Chighlight%28-4%29%2Chighlight%280%29%2C2%29%29

Step 4) Multiply by the inverse of the determinant



Multiply by 1%2Fd to get %281%2Fd%29%28matrix%282%2C2%2C1%2C-4%2C0%2C2%29%29

Plug in d=2 to get %281%2F2%29%28matrix%282%2C2%2C1%2C-4%2C0%2C2%29%29

Step 5) Multiply 1%2F2 by every element in the matrix (simplify and reduce if possible)



Multiply 1%2F2 by EVERY element to get

Multiply to get %28matrix%282%2C2%2C1%2F2%2C-4%2F2%2C0%2F2%2C2%2F2%29%29

Reduce each element: %28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29


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


Answer:

So the inverse of %28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29 is %28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29

This means that if A=%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29 then A%5E%28-1%29=%28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29




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Now let's use the inverse matrix to find the matrix X:


%28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29X=%28matrix%282%2C3%2C4%2C0%2C6%2C3%2C-1%2C5%29%29 Start with the given equation.


Left multiply both sides by the inverse %28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29



Multiply %28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29 and %28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29 to get %28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29 (this is the identity matrix). Let me know if you need help multiplying matrices.



%28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29X=%28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29 Multiply %28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29 and %28matrix%282%2C3%2C4%2C0%2C6%2C3%2C-1%2C5%29%29 to get %28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29.



X=%28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29 Multiply the identity matrix (which is I) by X to get I%2AX=X


So the answer is


X=%28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Can you help me solve the matrix equation.


Method 1:  Using the inverse:



Find the inverse of %28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29 

Do you know how to find the inverse of a 2x2 matrix?
If not post again asking how.  Or use method 2 below.

The inverse of %28matrix%282%2C2%2C2%2C4%2C0%2C1%29%29 is %28matrix%282%2C2%2C1%2F2%2C-2%2C0%2C1%29%29

Left-multiply both sides by this inverse:





matrix%281%2C3%2CX%2C+%22=%22%2C+%28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29%29

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

Method 2:  equating corresponding elements:



The right side is a 2x3 matrix.

The 2x2 matrix on the left side must be right 

multiplied by a 2x3 matrix to get a 2x3 matrix

Let X+=+%28matrix%282%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%29%29 be the
required 2x3 matrix. Then, substituting,

(matrix(2,3,a,b,c,d,e,f))







We equate corresponding elements on the 2nd row on both
sides:

d+=+3, e=-1, f=5

Substituting those:





We equate corresponding elements on the 1st row on both
sides:


So 

X+=+%28matrix%282%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%29%29

X+=+%28matrix%282%2C3%2C-4%2C2%2C-7%2C3%2C-1%2C5%29%29

Edwin