Example of Finding the Inverse of a 2x2 Matrix

This solution has been done by the calculator presented on the site.

It is necessary to calculate a matrix A-1, inverse to the given one:
A = 1 1
-1 3
Formula for calculating the inverse matrix:
A-1 = 1 / det A * A11 A21
A12 A22
A11 ... A22  are numbers (algebraic additions) that will be calculated later.
It is impossible to divide by zero. Therefore, if the determinant of A is zero, then it is impossible to calculate inverse matrix.
Let's calculate the determinant A.
det A = 1 1
-1 3
= 1 * 3 - 1 * ( -1) = 3 + 1 = 4
Determinant A is not zero. It is possible to calculate inverse matrix.
Let's calculate numbers (algebraic additions)   A11 ... A22
1 1
-1 3
Row number 1
Column number 1
Row 1 and column 1
have been deleted
A11 = ( -1) 1 + 1 * 3 = 3
1 1
-1 3
Row number 1
Column number 2
Row 1 and column 2
have been deleted
A12 = ( -1) 1 + 2 * -1 = 1
1 1
-1 3
Row number 2
Column number 1
Row 2 and column 1
have been deleted
A21 = ( -1) 2 + 1 * 1 = -1
1 1
-1 3
Row number 2
Column number 2
Row 2 and column 2
have been deleted
A22 = ( -1) 2 + 2 * 1 = 1
Result:
A-1 = 1 / det A * A11 A21
A12 A22
A-1 = 1 / 4 * 3 -1
1 1
A-1 = 3/4 -1/4
1/4 1/4
It is necessary to check that   A-1 * A = E.
We will use the penultimate form of the inverse matrix A-1. This will allow us to count without fractions.
3 -1
1 1
*
1 1
-1 3
=
b11 b12
b21 b22
b11 = 3 * 1 + ( -1) * ( -1) = 3 + 1 = 4
3 -1
1 1
*
1 1
-1 3
=
4 b12
b21 b22
b12 = 3 * 1 + ( -1) * 3 = 3 - 3 = 0
3 -1
1 1
*
1 1
-1 3
=
4 0
b21 b22
b21 = 1 * 1 + 1 * ( -1) = 1 - 1 = 0
3 -1
1 1
*
1 1
-1 3
=
4 0
0 b22
b22 = 1 * 1 + 1 * 3 = 1 + 3 = 4
3 -1
1 1
*
1 1
-1 3
=
4 0
0 4
It is necessary to multiply the result by 1/4
1/4 * 4 0
0 4
=
1 0
0 1
= E
Thus, the found matrix A-1 is inverse for the given matrix A.
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