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[n\'crt each of the following matrices. if possible:

Find Ihe inverse. if it exists. of each o f the following:

Find Ihe inverse. if il exists. of each of the following:

III Ererci.rex J3 (lml /4, I,ml'/' 111111 elleh gil'en IIlmrix A i~' IWII.ri/J!;II/ar (lII(/ II'rile;1 (/J {/ prodllcf oj elemel/fary matrices. (Him: First, wrile the il!l'er.l'11 (IS 1/ I!mduL'l of elemell/ary ma{rices: Ihell /l.I'e '{"/worelll 2.7.)

III Ererci.rex J3 (lml /4, I,ml'/' 111111 elleh gil'en IIlmrix A i~' IWII.ri/J!;II/ar (lII(/ II'rile;1 (/J {/ prodllcf oj elemel/fary matrices. (Him: First, wrile the il!l'er.l'11 (IS 1/ I!mduL'l of elemell/ary ma{rices: Ihell /l.I'e '{"/worelll 2.7.)

If A is a nonsingular malrix whose inverse is [~ ~l find A .

IfA '~ [: _I iJr."dA

Which or the following homogeneous systems luvc a no ntrivial solution'! (a) x +2y+3z=O 2),+ 2:=0 x + 2)' + 3: =0 (b) 2x+ y- ;:= 0 x - 2)' - 3;:=0 -3.l - )'+ 2:= 0 (c) 3.\'+ ),+3;:_0 - 2.r +2)' - 4:= 0 2.r-3),+5:= O

Which of thc following homogeneous systems have a nontrivial solution'] (a) x+)'+2:=0 2,r+y+ :=0 3x -)'+:=0 (b) .r - y + ;: = 0 2,'+ Y = 0 h - 2y+2z= O (c) 2\' - y + 5;: = 0 3x+ 2,,-3;:=0 .r- ), +4:=0

Find all value(s) of a for which the inverse o f I 0] o 0 2 " exists, What is A- I

For what values of (I docs the homogeneous system (I- I)X+ 2),=0 2.l + (1/ - 1, = 0 have a nontrivinl solution?

Provc Ihat is nonsingular if und only if (ld - be", 0, If this condition holds. show thut [ d ad - bc A- I = _{' (Id -I)c - b ] atl - be " . (/(1 - be

Let J o 2 26. 27. Let A be an m X II matrix. Show that A is row equil'alent toOifandonlyifA=O. Let A and B be III x II matrices. Show th;!t A is row equivalent to B if and only if AT is column equivalent to .'. Find the elementary matrix that as a postmultiplier of A performs the following elementary column operations on k

Prove that two III x II matrices A and B are row equivalent if and onl y if there exists a nonsingular matrix I' such that B = P A. (Him: Use Theorems 2.6 and 2.8.)

Let A and B be row equivalelllll XI! matrices. Prove th;!t A is nonsingular if and only if B is nonsingular.

t:l A <1m] B be II X I! m

Let A be an m X II matrix. Show that A is row equil'alent toOifandonlyifA=O.

Let A and B be III x II matrices. Show th;!t A is row equivalent to B if and only if AT is column equivalent to .

Show that a square matrix which has a row or a column consisting entirely of zeros must be singular.

(a) [s (A + B)- I = A - I + B- 1?

If A is an /I x II matrix . prove that A is nonsingular if and only if the linear system Ax = b has a unique solution for every /I x I matrix b.

Prove that the inverse of a nonsingular upper (lower) triangular matrix is upper (lower) triangular.

If the software you use has a command for computing reduced row echelon form. use it to determine whether the matrices A in Exercises 9, 10. and II have an inverse by operating on the matrix [A i I" J. (See Example 4.)

Repeat Exercise 32 on the matrices given in Exercise 63 of Section 1.5.