On the Quasi-inverse of a Non-square Matrix: An Infinite Solution

Applied Mathematical Sciences, Vol 11, 2017, no 27, 1337-1351 HIKARI Ltd, wwwm-hikaicom https://doiog/1012988/ams20177273 On the Quasi-invese of a Non-squae Matix: An Infinite Solution Ruben D Codeo J Math Depatment College of Ats and Sciences Univesity of Mindanao Bolton st, Davao City, Philippines Melvin B Manayon Math Depatment College of Ats and Sciences Univesity of Mindanao Bolton st, Davao City, Philippines Alben P Sagpang College of Teache Education Univesity of Mindanao Bolton st, Davao City, Philippines Copyight c 2017 Ruben D Codeo J, Melvin B Manayon and Alben P Sagpang This aticle is distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited Abstact A non-squae matix A is quasi-invetible if thee exists A diffeent fom the pseudo-invese matix such that eithe AA =I o A A = I such that A povides a solution to the linea system AX = B We descibe A as the quasi-invese of A This pape pesents the necessay and sufficient conditions fo the infinitude in the existence of quasi-inveses of a non-squae matix It also exhibits the popeties of a quasi-invese matix Moeove, this povides an altenative way of solving linea system with many solutions by employing the quasi-invese of a matix

1338 Ruben D Codeo J, Melvin B Manayon and Alben P Sagpang Mathematics Subject Classification: 15A09 Keywods: quasi-invetible matix, quasi-invese matix, linea system 1 Intoduction Conside solving a linea system AX = B when the numbe of unknowns exceeds the numbe of equations o the numbe of equations exceeds the numbe of unknowns Fo instance, if the numbe of unknowns is one moe than the numbe of equations, then the system has infinitely many solutions if they exist On the othe hand, if the numbe of equations is one moe than the numbe of unknowns then the system has a unique solution if it exists In dealing with these poblems we usually employ the taditional elimination method Now, the solution of the system can be expessed as X = A B whee A is the invese of A But note that the coefficient matix of the linea system given above is non-squae, so maybe by a mee coincident o by an accident, we ae actually computing the invese of a non-squae matix Since the solution of the linea system mentioned above need not be unique, the matix that satisfies the condition need not also to be unique This enlightens the way to evisit the discussion of finding the invese of a non-squae matix Not so long ago, the study of finding the invese of a non-squae matix has become an inteest of mathematicians such as Mooe[3] in 1920 and Stoe[6] in 2002 This gives ise to the concept of pseudo-invese matix which was fist exploed by EH Mooe[3] A pseudo-invese A + of matix A is a genealization of the invese matix, widely known as the Mooe-Penose pseudo-invese, which was independently descibed by E H Mooe, Ane Biehamma[1] in 1951 and Roge Penose[4] in 1955 Since the matix we conside is not necessaily squae, the pseudo-invese matix is one-sided That is, A + is a ight invese of an m n matix A if m < n and left invese othewise When efeing to a matix, the tem pseudo-invese, without futhe specification, is often used to indicate the Mooe-Penose pseudo-invese Even though the pseudo-invese is widely accepted as an invese of a nonsquae matix, it does not povide a solution to the linea system AX = B when matix A is of ode m n when m < n because if A + is the ight invese of A then we have AA + X = BA + which yields IX = BA + implies that X = BA + but BA + is not defined Due to this shotcoming, the eseaches intoduced a diffeent one-sided invese of a non-squae matix called the quasi-invese matix in ode to develop a new method of solving a linea system whose coefficient matix is non-squae In this case, the non-squae matix whose quasi-invese exists is descibed as quasi-invetible matix Fo the basic teminologies not stated hee, the eade is advised to efe to the book of Kolman and Hill[2]

On the quasi-invese of a non-squae matix: an infinite solution 1339 2 Basic Concepts Definition 21 A linea system is a collection of m linea equations with n unknowns x 1, x,, x n, a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m Definition 22 An m n matix A = [a ij ] of ode mn is a ectangula aay of numbes having m ows and n columns a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn The ow vectos of A ae (a 11 a 12 a 1n ), (a 21 a 22 a 2n ),, (a m1 a m2 a mn ) a 11 a 12 a 1n a 21 while the column vectos ae, a 22,, a 2n If m = n, then we a m1 a m2, a mn say that A is a squae matix of ode n If m n, we say that A is a nonsquae matix Moeove, if the ow vectos of A ae independent then we say that A has independent ows The independent columns of A ae defined analogously Definition 23 If A is an m n matix, then the matix B = [a pk ] obtained by deleting a ow/s o column/s o both is called a submatix of A Definition 24 Let A be an m n matix whee n m + 2 The m (m + 1) matix a 11 a 12 a 1m a 1p a 21 a 21 a 2m a 2p B = a m1 a m2 a mm a mp whee the (m + 1) th column enties a ip = n t=m+1 a it is called the diminished matix of A

1340 Ruben D Codeo J, Melvin B Manayon and Alben P Sagpang Definition 25 If A = [a ij ] is an m n matix, then the n m matix A T = [a T ij] whee a T ij = a ji is called the tanspose of A Definition 26 Let S = 1, 2,, n} aanged in ascending ode A eaangement j 1 j 2 j n of the elements of S is called a pemutation of S and is denoted by S n The numbe of pemutations of S is given by n! A pemutation j 1 j 2 j n is said to have an invesion if lage intege j pecedes a smalle one j s A pemutation is called even o odd accoding to whethe the total numbe of invesions is even o odd It should be noted that thee ae n!/2 even as well as odd pemutations Definition 27 Let A be an n n matix The deteminant of A, witten A is given by A = (±)a 1j1 a 2j2 a njn whee the summation anges ove all pemutations j 1 j 2 j n of the set S = 1, 2,, n} The sign + o depends on whethe the pemutation j 1 j 2 j n is even o odd Theoem 28 Let A and B be m n matices The following hold: i A = A T ii AB = A B Definition 29 Let A be an n n matix Let M ij be the (n 1) (n 1) submatix of A obtained by deleting the ith ow and jth column of A The deteminant M ij is called the mino of a ij The cofacto A ij of a ij is defined as A ij = ( 1) i+j M ij Definition 210 Let A be a squae matix of ode n The adjoint of A, denoted by adja is an n n matix whose i, jth element is the cofacto A ji of a ji That is, A 11 A 21 A n1 A 12 A 22 A n2 adja = A 1n A 2n A nn Theoem 211 If A is an n n matix and A 0, then A is invetible and A 1 = 1 A (adja) A 11 A 21 A A A 12 A 22 A A = A 1n A 2n A A A n1 A A n2 A A nn A

On the quasi-invese of a non-squae matix: an infinite solution 1341 Theoem 212 Let A be an n n matix Then A is nonsingula if and only if the linea system AX = B has a unique solution fo evey n 1 matix B The solution is given by X = A 1 B Definition 213 [5] Let A be an m n matix If A T A is invetible then, the left pseudo-invese A + L of A is defined as A + L = (AT A) 1 A T and A + LA = I o E If AA T is invetible then, the ight pseudo-invese A + R A + R = AT (AA T ) 1 and AA + R = I o E Remak 214 We use the notation A + to mean A + R o A+ L is defined as Theoem 215 [6] The following ae the popeties of a pseudo-invese matix A + of matix A ii If A is invetible, then A + = A 1 iii The pseudo-invese of a zeo matix is its tanspose iv (A + ) + = A v (αa) + = α 1 A +, α R, α 0 Definition 216 Let A be an m n matix The one-sided invese A, not necessaily equal to A +, such that AA = I o A A = I whee A can povide non-unique solutions to the linea system AX = B is called a quasi-invese of A The solution is given by X = A B We descibe matix A as quasiinvetible 3 Quasi-invese of a Non-squae Matix Lemma 31 Let A be an invetible matix and let A be a matix obtained fom A by adding a ightmost column o a bottom ow Then A is a quasiinvetible and thee exists matix B with a ow of such that eithe AB = I o BA = I a 11 a 12 a 1n Poof : Let A a 21 a 22 a 2n = be an invetible matix and let a n1 a n2 a nn a 11 a 12 a 1n a n+1 a 21 a 22 a 2n a 2(n+1) A = o A = a n1 a n2 a nn a n(n+1) a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn a (n+1)1 a (n+1)2 a (n+1)n

1342 Ruben D Codeo J, Melvin B Manayon and Alben P Sagpang Then A is an n m o an m n matix whee m = n+1 Since A is invetible, A 1 exists and by Theoem 211, A 1 = 1 (adja ) A 11 A 21 A n1 A 12 A 22 A n2 = A 1n A 2n A A nn [ ] A 1 Conside an m n o n m matix B = o B = [ A 0 1 0 ] If a 11 a 12 a 13 a 1m a 21 a 22 a 23 a 2m [ ] A A = and B = 1, then AB = I If A = 0 a n1 a n2 a n3 a nm a 11 a 12 a 1n a 21 a 22 a 2n and B = [ A 1 0 ], then BA = I Thus, B satisfies a m1 a m2 a mn the equiement as a quasi-invese of A Since B consists of a ow o column of zeos, we can eplace this ow o column by any eal numbe and the effect is that the enties A ji of A 1 will be added o subtacted by d ji That is, we can have A 11 ±d 11 A 21 ±d 21 A n1 ±d n1 o B = B = A 12 ±d 12 A 1n ±d 1n A 11 ±d 11 A 12 ±d 12 A 1n ±d 1n A 22 ±d 22 A 2n ±d 2n A n2 ±d n2 A 21 ±d 21 A 22 ±d 22 A 2n ±d 2n A nn ±d nn A n1 ±d n1 A n2 ±d n2 A nn ±d nn and we can obtain AB = I o BA = I Theefoe, A is quasi-invetible

On the quasi-invese of a non-squae matix: an infinite solution 1343 Theoem 32 Let A be an m n quasi-invetible matix whee m = n 1 and let A j be a squae submatix of A whose ode is m and is obtained by deleting the j th column of A Then the quasi-invese of A is an n m matix A = [b ji ] such that b ji = Aij + A j, if j is odd A ij A j, if j is even when the numbe of ows is even and Aij A j A b ji = n, if j is odd A ij + A j, if j is even when the numbe of ows is odd whee in both cases, 1 j m and b ni = fo i = 1, 2,, m, A ij is the cofacto of a ij of the invetible submatix A n obtained fom A by deleting the n th column That is, A = A 11 ± A 1 A 12 ± A 2 A 21 ± A 1 A 22 ± A 2 A m1 ± A 1 A m2 ± A 2 A 2m ± A 1m ± A mm ± Poof : Let A be an m n Since A is quasi-invetible, by Lemma 31, thee exists a matix B with a ow of such that AB = I Let A j be a squae submatix of A of ode m obtained by deleting the j th column of A Then A n is a submatix of A obtained by deleting the nth column which is invetible, and it follows that A n 0 So, let A = [b ji ] be an n m matix whee b ni = fo all i = 1, 2,, m such that AA = I Then we have b a 11 a 12 a 11 b 12 b 1m 1 0 0 1n AA a 21 a 22 a 2n b 21 b 22 b 2m 0 1 0 = = b a m1 a m2 a m1 b m2 b mm 0 0 1 0 mn 0 0 1 Simplifying, we obtain the linea system a 11 b 11 + a 12 b 21 + + a 1m b m1 + a 1n = 1 a 11 b 12 + a 12 b 22 + + a 1m b m2 + a 1n = 0 a 11 b 1m + a 12 b 2m + + a 1m b mm + a 1n = 0; a 21 b 11 + a 22 b 21 + + a 2m b m1 + a 2n = 0

1344 Ruben D Codeo J, Melvin B Manayon and Alben P Sagpang a 21 b 12 + a 22 b 22 + + a 2m b m2 + a 2n = 1 a 21 b 1m + a 22 b 2m + + a 2m b mm + a 2n = 0; a m1 b 11 + a m2 b 21 + + a mm b m1 + a mn = 0 a m1 b 12 + a m2 b 22 + + a mm b m12 + a mn = 0 a m1 b 1m + a m2 b 2m + + a mm b mm + a mn = 1 We pove hee the case when the numbe of ows is even and the case when the numbe of ows is odd can be poved similaly Solving fo b ji by elimination, we obtain (±)a2k2 a 3k3 a mkm +( (±)a 1k2 a 2k3 a nkn ) b 11 = (±)a1k1 a 2k2 a mkm (whee the summation anges ove all pemutations of k 2, k 3,, k m of the set 2, 3,, m}, k 2, k 3,, k n of the set 2, 3,, n}) and k 1, k 2,, k m of the set 1, 2,, m} definition 26) = ( 1)2 M 11 + A 1 = A 11 + A 1, (definition 27 and 29) b 12 = (±)a1k2 a 3k3 a mkm +( (±)a 2k2 a 2k3 a nkn ) (±)a1k1 a 2k2 a mkm all pemutations of k 2, k 3,, k m of the set 2, 3,, m}) = ( 1)3 M 21 + A 1 = A 21 + A 1 b 1m = (±)a2k2 a 2k2 a m 1km +( (±)a 2k2 a 2k3 a nkn ) (±)a1k1 a 2k2 a mkm (whee the summation anges ove ove all pemutations of k 2, k 3,, k m of the set 2, 3,, m}) = ( 1)m+1 M m1 + A 1 = A m1 + A 1, (±)a1k2 a 3k3 a mkm ( (±)a 1k1 a 2k3 a nkn ) b 21 = (±)a1k1 a 2k2 a mkm (whee the summation anges (whee the summation anges ove all pemutations of k 2, k 3,, k m of the set 2, 3,, m} and k 1, k 3,, k n of the set 1, 3,, n} = ( 1)3 M 12 A 2 = A 12 A 2, (±)a1k1 a 2k3 a mkm ( (±)a 1k1 a 2k3 a nkn ) b 22 = (±)a1k1 a 2k2 a mkm all pemutations of k 1, k 2,, k m of the set 1, 3,, m}) = ( 1)4 M 22 A 2 = A 22 A 2, b 2m = (±)a1k1 a 3k2 a mkm ( (±)a 1k1 a 2k3 a nkn ) (±)a1k1 a 2k2 a mkm all pemutations k 1, k 3, k m of the set 1, 2,, m}) = ( 1)m+2 M m2 A 2 = A m2 A 2, (whee the summation anges ove (whee the summation anges ove

On the quasi-invese of a non-squae matix: an infinite solution 1345 b m1 = (±)a2k1 a 2k2 a mkm 1 +( (±)a 1k1 a 2k2 a m 1kn 1 a mkn ) (±)a1k1 a 2k2 a mkn 1 a mkn (whee the summation anges ove all pemutations k 1, k 2,, k m 1 of the set 1, 2, 3,, m 1} and k 1, k 2,, k m of the set 1, 2,, n 1, n} = ( 1)1+m M 1m + = A 1m + (±)a1k1 a 2k3 a mkm 1 a mkn +( (±)a 1k1 a 2k2 a m 1kn 1 a mkn ) b m2 = (±)a1k1 a 2k2 a n 1kn 1 (whee the summation anges ove all pemutations of k 1, k 3,, k m of the set 1, 3,, n 2, n}) = ( 1)2+m M 2m + = A 2m +, b mm = (±)a1k1 a 2k2 a m 1km 1 +( (±)a 1k1 a 2k2 a m 1kn 1 am kn ) (±)a1k1 a 2k2 a mkm (whee the summation anges ove all pemutations k 1, k 2,, k m of the set 1, 2,, m}) = ( 1)m+m M mm + = Amm + Am Coollay 33 If A is an m n quasi-invetible matix with m = n + 1, then the quasi-invese is an n m matix A = [b ji ] fo 1 j n whee Aij + A i A b ji = m, if i is odd A ij A i, if i is even when the numbe of columns is even and Aij A i A b ji = m, if i is odd A ij + A i, if i is even when the numbe columns is odd b mi =, A i is a squae submatix of A obtained by deleting the i th ow, A ij is the cofacto of a ij of the invetible submatix A m obtained fom A by deleting the m th ow That is, A 11 ± A 1 A 12 ± A 2 A 1n ± A 21 ± A 1 A 22 ± A 2 A A A = m 2n ± A n1 ± A 1 A n2 ± A 2 A nn ± a 11 a 12 a 1n a 21 a 22 a 2n Poof : Suppose that A = Since A is quasi-invetible, a m1 a m2 a mn by Lemma 31, thee exists matix B such that BA = I Let

1346 Ruben D Codeo J, Melvin B Manayon and Alben P Sagpang b 11 b 12 b 1n a 11 a 21 a m1 b 21 b 22 b 2n B = Conside now a 12 a 22 a m2 AT = and b n1 b n2 b nn a 1n a 2n a mn b 11 b 21 b n1 b 12 b 22 b n2 B T = Since AT BT = (BA) T = IT = I, AT is quasiinvetible and its quasi-invese is B T Note that A T is an n m matix with b 1n b 2n b nn n = m 1 So the ode of A T satisfies the hypothesis of Theoem 32 Thus, by Theoem 32, we obtain B T = A 11 ± A 1 A 12 ± A 2 A 21 ± A 1 A 22 ± A 2 A n1 ± A 1 A n2 ± A 2 A 2m ± A 1n ± A nn ± whee b ij = Aji + A i, if i is odd A ji A i, if i is even when the numbe of columns is even and Aji A i A b ij = m, if i is odd A ji + A i, if i is even when the numbe of columns is odd Since (B T ) T = B, we have b T ij = b ji = Aij + A i, if i is odd A ij A i, if i is even when the numbe of columns is even and Aij A i b T A ij = b ji = m, if i is odd A ij + A i, if i is even when the numbe of columns is odd

On the quasi-invese of a non-squae matix: an infinite solution 1347 Let B = A Then A = A 11 ± A 1 A 12 ± A 1 A 1n ± A 1 A 21 ± A 2 A 22 ± A 2 A n2 ± A 2 A n1 ± A n2 ± A nn ± Coollay 34 Let A be an m n matix with n m + 2 If the diminished matix of A is quasi-invetible then A is quasi-invetible The quasi-invese of A is given by A = [b ji ] whee b ji = 0 o 1 fo m + 1 j n obtained by extending A as defined in Theoem 32 Poof : Let A be an m n matix with n m + 2 and suppose that the diminished matix B of A is quasi-invetible By Theoem 32, thee exists B such that BB = I Now, extend B to an n m matix A = [b ji ] such that b ji = 0 o 1 fo m + 1 j n Then we have AA = I Theoem 35 Let A be an m n quasi-invetible matix If A is ight invetible with A a ight invese then A T is left invetible and A T is a left invese of A T Similaly, if A is left invetible then A T is ight invetible and A T is a ight invese of A T Poof : Let A be an m n quasi-invetible matix Suppose that A is ight invetible By Lemma 31 and Theoem 32 thee exists A such that AA = I Now, A T A T = (AA ) T = I T = I which shows that A T is left invetible and A T is the left invese If A is left invetible, then thee exists A such that A A = I Now, A T A T = (A A) T = I T = I which shows that A T is ight invetible and A T is the ight invese Theoem 36 An m n quasi-invetible matix has infinitely many quasiinveses if and only if n = m 1 o n = m + 1 Poof : Let A be an m n matix Since A is quasi-invetible, by Theoem 32 and Lemma 31, thee exists an n m matix A such that AA = I o A A = I Note that thee ae n m o m n ows of depending on whethe n > m o n < m Suppose that thee ae infinitely many quasi-nveses of A Assume that n m 1 o n m + 1 If n < m 1 o n > m + 1 then by Coollay 34, the ows b ji = fo a quasi-invese matix hold only fo = 1 o = 0 which means that thee ae only two quasi-inveses of A contadicting the assumption Thus, n = m 1 o n = m + 1 Convesely, suppose that n = m 1 o n = m + 1 fo an m n matix A Then the numbe of ows o columns of A exceeds only by one By Theoem 32, the ow b ni = of a quasi-invese A assumes any eal numbe Let M = A : AA = I o A A = I} We have to show that M = + Suppose

1348 Ruben D Codeo J, Melvin B Manayon and Alben P Sagpang that M < + Then fo any >,, R such that b ni = of A does not yield a quasi-invese That is, fo any >, AA I which is a contadiction Since fo evey R if b ni = is the last ow A, we always have AA = I o A A = I, theefoe we conclude that thee ae infinitely many quasi-inveses of A Remak 37 Let A be the quasi-invese of an m n matix A Then AA = I m if m < n and A A = I n if m > n Remak 38 Let A be an m n quasi-invetible matix and A = [b ji ] whee b ni = fo all i = 1, 2,, m a quasi-invese of A If = 0, we call A the basic quasi-invese of A Theoem 39 Let A be an m n quasi-invetible matix and let B be an invetible matix of ode m if m < n and of ode n if m > n Then (BA) = A B 1 if m < n and (AB) = B 1 A if m > n Poof : Let A be an m n quasi-invetible matix Conside the following cases: Case 1 m < n Let B be an invetible matix of ode m Then BA is defined Since A is quasi-invetible, A exists and AA = I Now, (BA)(A B 1 ) = B(AA )B 1 = BIB 1 = BB 1 = I implies that A B 1 = (BA) Case 2 m > n Let B be an invetible matix of ode n Then AB is defined Since A is quasi-invetible, thee exists A such that A A = I Now, (B 1 A )(AB) = B 1 (A A)B = B 1 IB = B 1 B = I implies that B 1 A = (AB) Theoem 310 Let A be a matix If A is invetible, then A = A 1 and hence A = A +

On the quasi-invese of a non-squae matix: an infinite solution 1349 Poof : Let A be a matix Since A is invetible, A is a squae matix Hence, A has no ow of R Consequently, A 11 A 21 A n1 A A A A 12 A 22 A n2 A A A A = A 1n A 2n A nn A A A Thus, A = A 1 By Theoem 215, A 1 = A +, theefoe, A = A + Theoem 311 If A is a quasi-invetible matix then (ca) = c 1 A, 0 c R whee last ow of (ca) consists of c 1 Poof : Suppose that A is a quasi-invetible matix Then A is eithe left o ight invetible If A is ight invetible, then thee exists A such that AA = I Let 0 c R Then ca is quasi-invetible Now, (ca)(ca) = (ca)(a c 1 ) = c(aa c 1 ) = (cic 1 ) = cc 1 I = I which holds only if the last ow of (ca) consists of c 1 If A is left invetible, then thee exists A such that A A = I and ca is quasi-invetible Now, (ca) (ca) = (A c 1 )(ca) = A (cc 1 )A = A A = I Thus, in eithe case, (ca) = c 1 A 4 The Linea System AX = B This section discusses some esults of finding the solution of the linea system AX = B using the quasi-invese as well as the pseudo-invese of an m n matix A We conside the cases when m = n 1 o m = n + 1 Theoem 41 Let A be an m n matix with m = n 1, X = [x 1, x 2,, x n ] T, and B = [c 1, c 2,, c m ] T If A is quasi-invetible and x n = (c 1 +c 2 + +c m ), then the linea system AX = B has many solutions Each solution is given by X = A B Poof : Let A be anm n matix Then AX = B is a 11 a 12 a 1n x 1 c 1 a 22 a 23 a 2n x 2 = c 2 a m1 a m2 a mn x n c m Simplifying the left side and equating to the ight side, we obtain the linea system a 11 x 1 + a 12 x 2 + + a 1n x n = c 1 a 21 x 1 + a 22 x 2 + + a 2n x n = c 2

1350 Ruben D Codeo J, Melvin B Manayon and Alben P Sagpang a m1 x 1 + a m2 x 2 + + a mn x n = c m Solving each vaiable by elimination and substituting x n = (c 1 +c 2 + +c m ), we obtain x 1 = A 11 ± A 1 c 1 + A 21 ± A 1 c 2 + + A m1 ± A 1 c m x 2 = A 12 ± A 2 c 1 + A 22 ± A 2 c 2 + + A m2 ± A 2 c m x n 1 = A 1m ± A n 1 c 1 + A 2m ± A n 1 c 2 + + Amm ± A n 1 c m whee A ij is the adjoint of a ij and A j ae submatices of A of ode m obtained by deleting the jth column of A Fom Theoem 32, we find that Aij + A j A b ji = n, if j is odd A ij A j, if j is even when the numbe of ows is even and Aij A j A b ji = n, if j is even A ij + A j, if j is odd when the numbe of ows is odd with 1 j n 1 Thus, x 1 = b 11 c 1 + b 12 c 2 + + b 1m c m x 2 = b 21 c 1 + b 22 c 2 + + b 2m c m x n 1 = b m1 c 1 + b m2 c 2 + + b mm c m x 1 x 2 Hence, X = = x n Theefoe, X = A B b 11 c 1 + b 12 c 2 + + b 1m c m b 21 c 1 + b 22 c 2 + + b 2m c m b m1 c 1 + b m2 c 2 + + b mm c m c 1 + c 2 + + c m b 11 b 12 b 1m b 21 b 22 b 2m = b m1 b m2 b mm Coollay 42 Let AX = B be a linea system with A an m n matix whee m = n + 1 If A is quasi-invetible and the augmented matix has a ow equivalent to zeo, then the system has a unique solution given by X = A B whee A is a basic quasi-invese of A Poof : Let AX = B be a linea system with A an m n matix Suppose that A is quasi-invetible Since m = n + 1, by Lemma 31 thee exists A such that A A = I Since [A : B] has a ow equivalent to zeo, we can make this to be the last ow Let A be the basic quasi-invese of A Then we have, A AX = A B IX = A B X = A B c 1 c 2 c m

On the quasi-invese of a non-squae matix: an infinite solution 1351 Theoem 43 Let A be an m n matix whee m = n+1 If A T A is invetible and the linea system AX = B has a unique solution, then the solution can be given by X = A + L B whee A + L is a left pseudo-invese of A Poof : Let A be an m n matix whee m = n + 1 Suppose that A T A is invetible, (A T A) 1 exists and since the linea system AX = B has a solution, we have AX = B A T AX = A T B Multiplying by (A T A) 1, we obtain (A T A) 1 (A T A)X = (A T A) 1 A T B IX = (A T A) 1 A T B = A + L B Refeences [1] A Bjehamma, Application of Calculus of Matices to Method of Least Squaes: with Special Refeences to Geodetic Calculations, Elande, 1951 [2] B Kolman and D R Hill, Elementay Linea Algeba with Applications, 9th edition, Peason Education Inc, 2008 [3] E Mooe, On the Recipocal of the Geneal Algebaic Matix, 1920 [4] R Penose, J A Todd, A Genealized Invese fo Matices, Mathematical Poceedings of the Cambidge Philosophical Society, 51 (1955), 406 https://doiog/101017/s0305004100030401 [5] P N Sabes, Linea Algebaic Equations, SVD, and the Pseudo-invese, 2001 [6] J Stoe and R Bulisch, Intoduction to Numeical Analysis, 3d edition, Belin, New Yok, 2002 https://doiog/101007/978-0-387-21738-3 Received: Mach 9, 2017; Published: May 15, 2017