Find the inverse of the matrix
[1 2 3 1]
The given matrix is [1 2 3 1].
To find the inverse of the matrix, we need to use the formula for a 2x2 matrix:
[A B]
[C D]
The inverse matrix is given by:
1/(AD - BC) * [D -B]
[-C A]
In this case, A=1, B=2, C=3, and D=1.
So the inverse matrix is:
1/(1(1) - 2(3)) * [1 -2]
[-3 1]
Simplifying further:
1/(-5) * [1 -2]
[-3 1]
The inverse of the given matrix is:
[-1/5 2/5]
[ 3/5 -1/5]
To find the inverse of a matrix, you can use the following formula:
If A is the given matrix, and A^-1 is its inverse matrix, then:
A * A^-1 = I
Here, I represents the identity matrix.
Now, let's find the inverse of the matrix you provided:
[1 2 3 1]
The given matrix has four columns, so we are looking for a 4x4 inverse matrix.
To begin, we will augment the given matrix with the identity matrix of the same size:
[1 2 3 1 | 1 0 0 0]
[.............|.............]
[.............|.............]
[.............|.............]
Perform row operations to transform the given matrix into the identity matrix.
First, divide the first row by 1:
[1 2 3 1 | 1 0 0 0]
[.............|.............]
[.............|.............]
[.............|.............]
Next, replace the second row with the second row minus twice the first row:
[1 2 3 1 | 1 0 0 0]
[0 -2 -3 -1 | 0 1 0 0]
[.............|.............]
[.............|.............]
Then, replace the third row with the third row minus three times the first row:
[1 2 3 1 | 1 0 0 0]
[0 -2 -3 -1 | 0 1 0 0]
[0 0 0 0 | 0 0 1 0]
[.............|.............]
Lastly, replace the fourth row with the fourth row minus the first row:
[1 2 3 1 | 1 0 0 0]
[0 -2 -3 -1 | 0 1 0 0]
[0 0 0 0 | 0 0 1 0]
[0 0 0 0 | 0 0 0 1]
Now, the augmented matrix has been transformed into the identity matrix. The right-hand side of the augmented matrix is the inverse of the given matrix.
Therefore, the inverse of the matrix:
[1 2 3 1]
is:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]