Find the inverse of each of the following matrices, if possible
[i 3
1+i -i]
We can only take inverses of square matrices.
I see 2 numbers in the first row, but 3 in the second row.
Are there supposed to be brackets?
Is it
i 3
(1+i) - i
??
yes. so
i 3
(1+i) (-i)
the i and 3 are separate as well
in general the inverse of
a b
c d
is
1/(ad - bc)
times
d -b
-c a
e.g.
5 4
2 2
inverse is
1/(10-8)
times
2 -4
-2 5
=
1 -2
-1 2.5
Apply this rule to your matrix
To find the inverse of a matrix, let's call it A, we need to solve the equation AX = I, where X is the unknown matrix and I is the identity matrix.
Given the matrix A:
[i 3
1+i -i]
We can set up the equation AX = I as follows:
[i 3
1+i -i] [x1 x2] = [1 0
0 1]
Multiplying the matrices on the left side, we have:
[i * x1 + 3 * (1+i) i * x2 + 3 * -i] = [1 0
0 1]
Simplifying, we get a system of equations:
i * x1 + 3 * (1 + i) = 1
i * x2 + 3 * -i = 0
We can solve these two equations to find the values of x1 and x2.
First, let's solve the equation i * x2 + 3 * -i = 0:
i * x2 - 3i = 0
We can factor out i:
i * (x2 - 3) = 0
For this equation to be true, either i = 0 or x2 - 3 = 0.
Since i is a complex number, we need to consider the other case, x2 - 3 = 0:
x2 = 3
Now, let's solve the equation i * x1 + 3 * (1 + i) = 1:
i * x1 + 3 + 3i = 1
Rearranging the equation:
i * x1 + 3i = 1 - 3
Combining like terms:
i * x1 + 3i = -2
Factoring out i:
i * (x1 + 3) = -2
For this equation to be true, either i = 0 or x1 + 3 = -2.
Since i is a complex number, we need to consider the other case, x1 + 3 = -2:
x1 = -2 - 3
x1 = -5
Therefore, the unknown matrix X is:
[x1 x2] = [-5 3]
So the inverse of the given matrix A is:
[-5 3]