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To see the intermediate steps in calculating and inverse, just enter the matrix
A from above, and press Y, while keeping the step-by-step option active in
the calculator’s CAS. Use the following:
[[ 1,2,3],[3,-2,1],[4,2,-1]] `Y
After going through the different steps, the solution returned is:
What the calculator showed was not exactly a Gauss-Jordan elimination with
full pivoting, but a way to calculate the inverse of a matrix by performing a
Gauss-Jordan elimination, without pivoting. This procedure for calculating
the inverse is based on the augmented matrix (A
aug
)
n
×
n
= [A
n
×
n
|I
n
×
n
].
The calculator showed you the steps up to the point in which the left-hand half
of the augmented matrix has been converted to a diagonal matrix. From
there, the final step is to divide each row by the corresponding main diagonal
pivot. In other words, the calculator has transformed (A
aug
)
n
×
n
= [A
n
×
n
|I
n
×
n
],
into [I |A
-1
].
Inverse matrices and determinants
Notice that all the elements in the inverse matrix calculated above are divided
by the value 56 or one of its factors (28, 7, 8, 4 or 1). If you calculate the
determinant of the matrix A, you get det(A) = 56.
We could write, A
-1
= C/det(A), where C is the matrix
.
8614
8137
880
−
−=C
The result (A
-1
)
n
×
n
= C
n
×
n
/det(A
n
×
n
), is a general result that applies to any non-
singular matrix A. A general form for the elements of C can be written based
on the Gauss-Jordan algorithm.
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