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Question

Find the matrix A such that
(i) 1101 A=335101

(ii) A 123456=-7-8-9 2 4 6

(iii) 413 A=-484-121-363

(iv) 213-10-1-11001110-1=A

(v) 2-1 10-34 A=-1-8-10 1-2 -5 922 15


(vi) A=123456=-7-8-9 2 4 6 11 10 9

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Solution

i Let A=xyzabc1101xyzabc=335101x+ay+bz+c0+a0+b0+c=335101x+ay+bz+cabc=335101The corresponding elements of two equal matrices are equal.x+a=3 ...1y+b=3 ...2 z+c=5 ...3 a=1, b=0 and c=1Putting the value of a in eq. 1, we get x+1=3x=3-1 x=2Putting the value of b in eq. 2, we get y+b=3y+0=3 y=3Putting the value of c in eq. 3, we get z+1=5z=5-1 z=4 A=234101

ii Let A=wxyzA123456=-7-8-9246wxyz123456=-7-8-9246w+4x2w+5x3w+6xy+4z2y+5z3y+6z=-7-8-9246The correspnding elements of two equal matrices are equal. 3w+6x=-9 ...1 y+4z=2 y=2-4z ...2 w+4x=-7 w=-7-4x ...3 2y+5z=4 ...4 Putting the value of w in eq. 1, we get3-7-4x+6x=-9-21-12x+6x=-9-6x=12x=-2Putting the value of x in eq. 3, we getw=-7-4-2 w=-7+8w=1Putting the value of y in eq. 4, we get22-4z+5z=44-8z+5z=44-3z=4-3z=0z=0Putting the value of z in eq. 2, we get y=2-40 y=2 A=1-220


iii Let A=xyz413xyz=-484-121-3634x4y4zxyz3x3y3z=-484-121-363The corresponding elements of two equal matrices are equal.4x=-4 ...14y=8 ...2 4z=4 ...3 x=-1, y=2 and z=1 A=-121


iv Let A=x213-10-1-11001110-1=A213-10-1-11001110-1=x-2-1+00+1+3-2+0+310-1=x-34110-1=x-3+0-1=x-4=xThe corresponding elements of two equal matrices are equal.x=-4 A=-4


v 2-1 10-34A=-1-8-10 1-2 -5 922 15Let A=xyzabc 2-1 10-34xyzabc=-1-8-10 1-2 -5 922 152x-a2y-b2z-cxyz-3x+4a-3y+4b-3z+4c=-1-8-10 1-2 -5 922 15By comparing the elements of second row, we getx=1, y=-2, z=-5By comparing the elements of first row, we get2x-a=-12-a=-1a=3Also,2y-b=-8-4-b=-8b=4And,2z-c=-10-10-c=-10c=0 A=1-2-5340

vi A123456=-7-8-9 2 4 6 11 10 9Let A=xaybzcxaybzc123456=-7-8-9 2 4 6 11 10 9x+4a2x+5a3x+6ay+4b2y+5b3y+6bz+4c2z+5c3z+6c=-7-8-9 2 4 6 11 10 9By comparing the corresponding elements, we getx+4a=-7 and 2x+5a=-8a=-2 and x=1Also,y+4b=2 and 2y+5b=4b=0 and y=2And,z+4c=11 and 2z+5c=10c=4 and z=-5 A=1-220-54

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