Find matrices X given an equation.

[candymountain]

Homework Statement

find all matrices x that satisfy the given matrix equation
[ 1 2 3
4 5 6] * X = I_2

I_2 is the identity matrix 2x2

Homework Equations

The Attempt at a Solution

I just inverted the square matrix
[ 1 2
4 5]
so it becomes
[5 -2
-4 1 ]
so X should be
[ 5 -2
-4 1
0 0]

but my book solution introduces 2 variables S and T to capture all the solutions, how do i do this?
*ps, how do i make my matrices neater?

 

[hunt_mat]

The way you went about it is a little odd. I personally would have written:
$$\left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right) \left( \begin{array}{cc} a_{1} & a_{2} \\ a_{3} & a_{4} \\ a_{5} & a_{6} \end{array} \right) =\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$$
Multiplied them out and solved the linear equations. You will get four equations to solve for 6 unknowns, this is where the variables t & s come into it.

 

[Raskolnikov]

I would just treat it as two simple linear algebra problems of finding the solution space for Ax=b:$$\left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right)$$

simplifies to the reduced row-echelon form matrix

$$\left( \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 2 \end{array} \right)$$

So we have
$$\left( \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 2 \end{array} \right) \left( \begin{array}{c} a_{1} \\ a_{3} \\ a_{5} \end{array} \right) =\left( \begin{array}{c} 1 \\ 0 \end{array} \right)$$

and
$$\left( \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 2 \end{array} \right) \left( \begin{array}{c} a_{2} \\ a_{4} \\ a_{6} \end{array} \right) =\left( \begin{array}{c} 0 \\ 1 \end{array} \right)$$

 

[candymountain]

thanks for the replies.
I see how it's more intuitive to do it the algebraic way, so I'll just discard the inverse trick.
Using the rref makes it pretty simple, but when i reduce the one on the left side, do I reduce it on the right as well, if say we're given a matrix other than an I_n ?

 

[Raskolnikov]

thanks for the replies.
I see how it's more intuitive to do it the algebraic way, so I'll just discard the inverse trick.
Using the rref makes it pretty simple, but when i reduce the one on the left side, do I reduce it on the right as well, if say we're given a matrix other than an I_n ?

Yea, sorry I forgot to change the right side. But yep, just apply Gauss-Jordan elimination to the augmented matrix (which includes the right-hand side).

 

[candymountain]

can anyone verify this?

my 1st column came out to be
a = t
b = -2 -2t
c = 5/3 +t

and 2nd column
d= s
e= 1-2s
f= -2/3+s