Linear Algebra The Inverse of a Matrix Dr. Bisher M. Iqelan biqelan@iugaza.edu.ps Department of Mathematics The Islamic University of Gaza 2017-2018, Semester 2 Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 1 / 12
Invertible Matrix Definition 1 An n n matrix A is invertible if there exists a n n matrix B so that AB = I n = BA. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 2 / 12
Invertible Matrix Definition 1 An n n matrix A is invertible if there exists a n n matrix B so that AB = I n = BA. 2 B is called the inverse of A and is denoted by A 1. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 2 / 12
Invertible Matrix Definition 1 An n n matrix A is invertible if there exists a n n matrix B so that AB = I n = BA. 2 B is called the inverse of A and is denoted by A 1. 3 A matrix which is not invertible is said to be singular. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 2 / 12
The Speical Case of 2 2 Matrices Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 3 / 12
The Speical Case of 2 2 Matrices Definition ( ) a b Let A =. We define (and this only works for 2 2 matrices) the c d determinant of A to be the quantity det(a) = ad bc. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 3 / 12
The Speical Case of 2 2 Matrices Definition ( ) a b Let A =. We define (and this only works for 2 2 matrices) the c d determinant of A to be the quantity Theorem ( a b Let A = c d which case det(a) = ad bc. ). Then A is invertible if and only if det(a) is non-zero, in A 1 = 1 det(a) If det(a) = 0 then A is singular. ( d b c a ). Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 3 / 12
Theorem If A is an invertible m m matrix, then for every b R n, the equation A x = b has a unique solution, namely x = A 1 b. Proof. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 4 / 12
Theorem If A is an invertible m m matrix, then for every b R n, the equation A x = b has a unique solution, namely x = A 1 b. Proof. Theorem 1 If A is invertible, then so is A 1, and (A 1 ) 1 = A. 2 If A and B are invertible n n matrices then so is AB, and (AB) 1 = B 1 A 1. 3 If A is invertible, then so is A T, and (A T ) 1 = (A 1 ) T. Proof. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 4 / 12
Elementary Row Operations Recall Denoting rows r and s by R r and R s, the row operations are: R r R s Interchange rows R r and R s of a matrix. cr r For a non-zero c R, replace R r by cr r. cr s + R r Replace R r by cr s + R r Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 5 / 12
Elementary Row Operations Recall Denoting rows r and s by R r and R s, the row operations are: R r R s Interchange rows R r and R s of a matrix. cr r For a non-zero c R, replace R r by cr r. cr s + R r Replace R r by cr s + R r Definition An elementary matrix is any n n matrix that can be obtained by performing a single elementary row operation to I n. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 5 / 12
Elementary Matrices Example We construct three elementary matrices below. 1 0 0 1 0 0 0 1 0 R 3+R 2 0 1 2 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 R 2 R 3 3R 1 1 0 0 0 0 1 0 1 0 3 0 0 0 1 0 0 0 1 Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 6 / 12
Example Multiply the general 3 3 matrix on the left by each of the above matrices. 1 0 0 0 1 2 0 0 1 1 0 0 0 0 1 0 1 0 3 0 0 0 1 0 0 0 1 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = = = Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 7 / 12
Exercise For a matrix having 4 rows, write down the elementary matrices which perform the following elementary row operations. 1 R 1 R 3 2 3R 2 3 7R 4 + R 2 Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 8 / 12
Exercise For a matrix having 4 rows, write down the elementary matrices which perform the following elementary row operations. 1 R 1 R 3 2 3R 2 3 7R 4 + R 2 Exercise Write down the inverse for each of the elementary matrices above. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 8 / 12
Note 1 If I n R E, then Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 9 / 12
Note 1 If I n R E, then for any matrix A with n rows, A R EA. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 9 / 12
Note 1 If I n R E, then for any matrix A with n rows, A R EA. 2 So, if A can be row reduced to B by a sequence of row operations R R 1, R 2,..., R k and I i n Ei, then Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 9 / 12
Note 1 If I n R E, then for any matrix A with n rows, A R EA. 2 So, if A can be row reduced to B by a sequence of row operations R R 1, R 2,..., R k and I i n Ei, then B = E k E k 1 E 2 E 1 A. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 9 / 12
Note 1 If I n R E, then for any matrix A with n rows, A R EA. 2 So, if A can be row reduced to B by a sequence of row operations R R 1, R 2,..., R k and I i n Ei, then B = E k E k 1 E 2 E 1 A. (i.e. A R 1 E 1 A Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 9 / 12
Note 1 If I n R E, then for any matrix A with n rows, A R EA. 2 So, if A can be row reduced to B by a sequence of row operations R R 1, R 2,..., R k and I i n Ei, then B = E k E k 1 E 2 E 1 A. (i.e. A R 1 E 1 A R 2 E 2 (E 1 A) Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 9 / 12
Note 1 If I n R E, then for any matrix A with n rows, A R EA. 2 So, if A can be row reduced to B by a sequence of row operations R R 1, R 2,..., R k and I i n Ei, then B = E k E k 1 E 2 E 1 A. (i.e. A R 1 E 1 A R 2 E 2 (E 1 A) R 3 E 3 E 2 E 1 A Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 9 / 12
Note 1 If I n R E, then for any matrix A with n rows, A R EA. 2 So, if A can be row reduced to B by a sequence of row operations R R 1, R 2,..., R k and I i n Ei, then B = E k E k 1 E 2 E 1 A. (i.e. A R 1 E 1 A R 2 E 2 (E 1 A) R 3 E 3 E 2 E 1 A Rk E k E k 1 E 2 E 1 A = B). Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 9 / 12
Note 1 If I n R E, then for any matrix A with n rows, A R EA. 2 So, if A can be row reduced to B by a sequence of row operations R R 1, R 2,..., R k and I i n Ei, then B = E k E k 1 E 2 E 1 A. (i.e. A R 1 E 1 A R 2 E 2 (E 1 A) R 3 E 3 E 2 E 1 A Rk E k E k 1 E 2 E 1 A = B). 3 Since each row operation is invertible, each elementary matrix is invertible. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 9 / 12
Theorem An n n matrix A is invertible if and only if A I n, in which case the sequence of elementary row operations which transform A to the identity also transform the identity matrix I n to A 1. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 10 / 12
Theorem An n n matrix A is invertible if and only if A I n, in which case the sequence of elementary row operations which transform A to the identity also transform the identity matrix I n to A 1. Note Thus if A R 1 R 2 Rk I n then Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 10 / 12
Theorem An n n matrix A is invertible if and only if A I n, in which case the sequence of elementary row operations which transform A to the identity also transform the identity matrix I n to A 1. Note Thus if A R 1 R 2 Rk I n then [A : I n ] R 1 R 2 Rk [I n : A 1 ]. Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 10 / 12
Example 1 2 1 Let A = 2 3 1. Find A 1. 3 5 1 Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 11 / 12
H.W HW Sec. 2.2, p111: # 1,4,5,7,9,10,13,14,31,35 Dr. Bisher M. Iqelan (IUG) Sec.2.2: The Inverse of a Matrix 2nd Semester, 2017-2018 12 / 12