SOLUTION: 1. if A= [ 2 -4 ] [ -2 5 ]find A ^-1 2. Solve for X given that [ -5 0 -1 ] [ 2 1 0 ] =4 [ x

Algebra ->  Matrices-and-determiminant -> SOLUTION: 1. if A= [ 2 -4 ] [ -2 5 ]find A ^-1 2. Solve for X given that [ -5 0 -1 ] [ 2 1 0 ] =4 [ x       Log On


   

Question 942989: 1. if A= [ 2 -4 ]
[ -2 5 ]find A ^-1
2. Solve for X given that [ -5 0 -1 ]
[ 2 1 0 ] =4
[ x 3 0 ]
3. 3x +2y =5
-x +5y = 7
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

1.
Solved by pluggable solver: Finding the Inverse of a 2x2 Matrix

To find the inverse of the matrix A=%28matrix%282%2C2%2C2%2C-4%2C-2%2C5%29%29, we can follow these steps:

Step 1) Find the determinant



The determinant of %28matrix%282%2C2%2C2%2C-4%2C-2%2C5%29%29 is abs%28matrix%282%2C2%2C2%2C-4%2C-2%2C5%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%2C-4%2C-2%2Chighlight%285%29%29%29 to get %28matrix%282%2C2%2Chighlight%285%29%2C-4%2C-2%2Chighlight%282%29%29%29

Step 3) Change the sign



Now change the sign of the highlighted values %28matrix%282%2C2%2C5%2Chighlight%28-4%29%2Chighlight%28-2%29%2C2%29%29 to get %28matrix%282%2C2%2C5%2Chighlight%284%29%2Chighlight%282%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%2C5%2C4%2C2%2C2%29%29

Plug in d=2 to get %281%2F2%29%28matrix%282%2C2%2C5%2C4%2C2%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%2C5%2F2%2C4%2F2%2C2%2F2%2C2%2F2%29%29

Reduce each element: %28matrix%282%2C2%2C5%2F2%2C2%2C1%2C1%29%29


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Answer:

So the inverse of %28matrix%282%2C2%2C2%2C-4%2C-2%2C5%29%29 is %28matrix%282%2C2%2C5%2F2%2C2%2C1%2C1%29%29

This means that if A=%28matrix%282%2C2%2C2%2C-4%2C-2%2C5%29%29 then A%5E%28-1%29=%28matrix%282%2C2%2C5%2F2%2C2%2C1%2C1%29%29



2.

%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29
the determinant is:%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%28di-fg%29%2Bc%28dh-eg%29
you have %28matrix%283%2C3%2C-5%2C0%2C-1%2C2%2C1%2C0%2Cx%2C3%2C0%29%29
so, a=-5, b=0, c=-1, d=2,e=1,f=0,g=x,h=3, and i=0
the determinant is equal to 4:
a%28ei-fh%29-b%28di-fg%29%2Bc%28dh-eg%29=4
-5%281%2A0-0%2A3%29-0%282%2A0-0%2Ax%29-1%282%2A3-1%2Ax%29=4
0-0-1%286-x%29=4
-6%2Bx=4
x=4%2B6
x=10
check determinant if x=10
Solved by pluggable solver: Finding the Determinant of a 3x3 Matrix

If you have the general 3x3 matrix:

%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29

the determinant is:

Which further breaks down to:



Note: abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29, abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29 and abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 are determinants themselves.
If you need help finding the determinant of 2x2 matrices (which is required to find the determinant of 3x3 matrices), check out this solver

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From the matrix %28matrix%283%2C3%2C-5%2C0%2C-1%2C2%2C1%2C0%2C10%2C3%2C0%29%29, we can see that a=-5, b=0, c=-1, d=2, e=1, f=0, g=10, h=3, and i=0

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

abs%28matrix%283%2C3%2C-5%2C0%2C-1%2C2%2C1%2C0%2C10%2C3%2C0%29%29=0-0%2B4 Multiply

abs%28matrix%283%2C3%2C-5%2C0%2C-1%2C2%2C1%2C0%2C10%2C3%2C0%29%29=4 Combine like terms.


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Answer:

So abs%28matrix%283%2C3%2C-5%2C0%2C-1%2C2%2C1%2C0%2C10%2C3%2C0%29%29=4, which means that the determinant of the matrix %28matrix%283%2C3%2C-5%2C0%2C-1%2C2%2C1%2C0%2C10%2C3%2C0%29%29 is 4


3.
3x+%2B2y+=5
-x+%2B5y+=+7

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax%2B2%2Ay=5
-1%2Ax%2B5%2Ay=7

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 3 and -1 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and -1 is -3, we need to multiply both sides of the top equation by -1 and multiply both sides of the bottom equation by -3 like this:

-1%2A%283%2Ax%2B2%2Ay%29=%285%29%2A-1 Multiply the top equation (both sides) by -1
-3%2A%28-1%2Ax%2B5%2Ay%29=%287%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
-3%2Ax-2%2Ay=-5
3%2Ax-15%2Ay=-21

Notice how -3 and 3 add to zero (ie -3%2B3=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-3%2Ax%2B3%2Ax%29-2%2Ay-15%2Ay%29=-5-21

%28-3%2B3%29%2Ax-2-15%29y=-5-21

cross%28-3%2B3%29%2Ax%2B%28-2-15%29%2Ay=-5-21 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-17%2Ay=-26

y=-26%2F-17 Divide both sides by -17 to solve for y



y=26%2F17 Reduce


Now plug this answer into the top equation 3%2Ax%2B2%2Ay=5 to solve for x

3%2Ax%2B2%2826%2F17%29=5 Plug in y=26%2F17


3%2Ax%2B52%2F17=5 Multiply



3%2Ax%2B52%2F17=5 Reduce



3%2Ax=5-52%2F17 Subtract 52%2F17 from both sides

3%2Ax=85%2F17-52%2F17 Make 5 into a fraction with a denominator of 17

3%2Ax=33%2F17 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%2833%2F17%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=11%2F17 Multiply the terms on the right side


So our answer is

x=11%2F17, y=26%2F17

which also looks like

(11%2F17, 26%2F17)

Notice if we graph the equations (if you need help with graphing, check out this solver)

3%2Ax%2B2%2Ay=5
-1%2Ax%2B5%2Ay=7

we get



graph of 3%2Ax%2B2%2Ay=5 (red) -1%2Ax%2B5%2Ay=7 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (11%2F17,26%2F17). This verifies our answer.