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3x3 matrix inverse
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1
math 140 - calculating the inverse of a 3! 3 matrix.
• Find the inverse of
A =
!
"1 "1 10 "2 1"2 "3 0
#
$
(A|I) =
!
"1 "1 1 1 0 00 "2 1 0 1 0"2 "3 0 0 0 1
#
$
2!R1+R3"R3"""""""""#
!
"1 "1 1 1 0 00 "2 1 0 1 00 "5 2 2 0 1
#
$ (#1/2)!R2"R2""""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 "5 2 2 0 1
#
$
5!R2+R3"R3"""""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 "1/2 2 "5/2 1
#
$ #2!R3"R3"""""""#
!
"1 "1 1 1 0 00 1 "1/2 0 "1/2 00 0 1 "4 5 "2
#
$
R2+(1/2)!R3"R2""""""""""""#
!
"1 "1 1 1 0 00 1 0 "2 2 "10 0 1 "4 5 "2
#
$ R1+(#1)!R3"R1"""""""""""#
!
"1 "1 0 5 "5 20 1 0 "2 2 "10 0 1 "4 5 "2
#
$
R1+R2"R1""""""""#
!
"1 0 0 3 "3 10 1 0 "2 2 "10 0 1 "4 5 "2
#
$
=$ A#1 =
!
"3 "3 1"2 2 "1"4 5 "2
#
$
1