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Inverting 2 ร 2 matrices
In this note we invert the general 2ร2 matrix as in Theorem 1.4.5 of Antonโ Rorres. However, we apply only the standard inversion method, with no guesswork or ingenuity needed.
Theorem 1 The 2 ร 2 matrix A =
[ a b c d
] is invertible if and only if โ 6 = 0, where
we write โ = ad โ bc. When โ 6 = 0, the inverse is
Aโ^1 =
[ d โb โc a
]
Proof We row reduce the 2ร4 partitioned matrix
[A|I] =
[ a b 1 0 c d 0 1
] (2)
to obtain the reduced row echelon matrix [I|Aโ^1 ]. There are two cases, depending on whether a = 0 or not.
Case a 6 = 0 We multiply row 1 by 1/a to get
[ 1 b/a 1 /a 0 c d 0 1
]
Then we subtract c times row 1 from row 2 to obtain [ 1 b/a 1 /a 0 0 โ/a โc/a 1
]
where we write the entry at row 2, column 2 as d โ bc/a = โ/a. If โ = 0, inversion breaks down at this point, as we will not get a leading 1 in column 2; otherwise, we multiply row 2 by a/โ to get the row echelon form
[ 1 b/a 1 /a 0 0 1 โc/โ a/โ
]
Finally, we subtract b/a times row 2 from row 1 to get the desired reduced row echelon matrix, whose right half we read off as Aโ^1 , in the required form, [ 1 0 d/โ โb/โ 0 1 โc/โ a/โ
]
where at row 1, column 3 we write
1 a
bc aโ
ad โ bc + bc aโ
d โ
110.201 Linear Algebra JMB File: twoinv, Revision A; 27 Aug 2001; Page 1
2 Inverting 2 ร 2 matrices
Case a = 0 We must have c 6 = 0 for inversion to progress, otherwise we have a column of zeros and will never get a leading 1 in column 1. First we switch rows 1 and 2 in (2), (^) [ c d 0 1 0 b 1 0
]
Now we multiply row 1 by 1/c to get the leading 1 in row 1, [ 1 d/c 0 1 /c 0 b 1 0
]
Again, inversion breaks down here unless b 6 = 0 because we need a leading 1 in column
- We multiply row 2 by 1/b to get the row echelon matrix [ 1 d/c 0 1 /c 0 1 1 /b 0
]
Finally, we subtract d/c times row 2 from row 1 to obtain the reduced row echelon matrix (^) [ 1 0 โd/bc 1 /c 0 1 1 /b 0
]
Because now โ = โbc, this is what we want. (The condition โ 6 = 0 is exactly what we need to guarantee that c 6 = 0 and b 6 = 0.) Because the expression โ occurs everywhere, it deserves a name.
Definition 3 The determinant det(A) of the 2ร2 matrix A is the expression
det(A) = โ = ad โ bc
The method generalizes in principle to produce a formula for the inverse of a general nรn matrix, so we know a formula exists. Even for n = 3, we need a better way to find it.
110.201 Linear Algebra JMB File: twoinv, Revision A; 27 Aug 2001; Page 2
