CHAPTER 2 MATRIX ALGEBRA
2.2 The Inverse of a Matrix
• Inverse
An n × n matrix A is invertible if there exists an n × n matrix C such that
AC = In = CA
If A is invertible, we call C is its inverse and denote it by A−1 .
1 −2 −5 2
• Example Given that A = and C = , find AC and CA.
3 −5 −3 1
Hence, find A−1 .
1
,2
• Example Determine when is a 2 × 2 matrix invertible.
• Note If a matrix is invertible, the inverse is unique.
, 3
6 −7
• Example Find the inverse of A = .
−8 9
6x1 − 7x2 = 1
• Example Solve the linear system .
− 8x + 9x = −2
1 2
• Theorem
If A is an invertible matrix, the linear system Ax = b has a unique solution.
1 −2 6 −7
• Example Given that A = , B = , find (AT )−1 and
3 −5 −8 9
(AB)−1 .
, 4
• Theorem
• (A−1 )−1 = A
• (AB)−1 = B −1 A−1
• (AT )−1 = (A−1 )T
• Elementary Matrices
An elementary matrix is one that is obtained by performing a single elemen-
tary row operation on an identity matrix.
2.2 The Inverse of a Matrix
• Inverse
An n × n matrix A is invertible if there exists an n × n matrix C such that
AC = In = CA
If A is invertible, we call C is its inverse and denote it by A−1 .
1 −2 −5 2
• Example Given that A = and C = , find AC and CA.
3 −5 −3 1
Hence, find A−1 .
1
,2
• Example Determine when is a 2 × 2 matrix invertible.
• Note If a matrix is invertible, the inverse is unique.
, 3
6 −7
• Example Find the inverse of A = .
−8 9
6x1 − 7x2 = 1
• Example Solve the linear system .
− 8x + 9x = −2
1 2
• Theorem
If A is an invertible matrix, the linear system Ax = b has a unique solution.
1 −2 6 −7
• Example Given that A = , B = , find (AT )−1 and
3 −5 −8 9
(AB)−1 .
, 4
• Theorem
• (A−1 )−1 = A
• (AB)−1 = B −1 A−1
• (AT )−1 = (A−1 )T
• Elementary Matrices
An elementary matrix is one that is obtained by performing a single elemen-
tary row operation on an identity matrix.
