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Question

Solve the following determinant equations:

(i) x+abcax+bcabx+c=0

(ii) x+axxxx+axxxx+a=0, a0

(iii) 3x-83333x-83333x-8=0

(iv) 1xx21aa21bb2=0, ab

(v) x+1352x+2523x+4=0

(vi) 1xx31bb31cc3=0, bc

(vii) 15-2x11-3x7-x111714101613=0

(viii) 11xp+1p+1p+x3x+1x+2=0

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Solution

(i)

Let =x+abcax+bcabx+c=x+a+b+cbcx+a+b+cx+bcx+a+b+cbx+c Applying C1C1+C2+C3=x+a+b+c1bc1x+bc1bx+c =x+a+b+c1bc0 x01bx+c Applying R2R2-R1=x+a+b+c1bc0 x000x Applying R3R3-R1=x+a+b+cx2-0=0 Givenx2=0 or x+a+b+c=0x=0 or x=-a+b+c

(ii)
Let =x+axxxx+axxxx+a=3x+axx3x+ax+ax3x+axx+a Applying C1C1+C2+C3=3x+a1xx1x+ax1xx+a=3x+a1xx0a01xx+a Applying R2R2-R1=3x+a1xx0a000a Applying R3R3-R1=3x+aa2-0=0x=-a3

(iii)

Let =3x-83333x-83333x-8=3x-2333x-23x-833x-233x-8 Applying C1=C1+C2+C3=3x-213313x-83133x-8 =3x-213303x-110133x-8 Applying R2R2-R1=3x-213303x-110003x-11 Applying R3R3-R1=3x-23x-112=0x=23,113,113

(iv)
Let =1xx21aa21bb2=1xx20x-ax2-a21bb2 Applying R2R1-R2=1xx20x-ax2-a20x-bx2-b2 Applying R3R1-R3=x-ax-b1xx201x+a01x+b =x-ax-bx+b-x-a=0x=a,b

(v)

Let =x+1352x+2523x+4=x+935x+9x+25x+93x+4 Applying C1=C1+C2+C3=x+91351x+2513x+4 =x+91350x-1013x+4 Applying R2R2-R1=x+91350x-1000x-1 Applying R3R3-R1=x+9x-12=0x=-9, 1, 1

(vi)
Let =1xx31bb31cc3=1xx30b-xb3-x31cc3 Applying R2R2-R1=1xx30b-xb3-x30c-xc3-x3 Applying R3R3-R1=1xx30x-bx3-b30x-cx3-c3=x-bx-c1xx201x2+xb+b201x2+xc+c2 =x-bx-cxc-b-b2+c2=0x=b, c, -b+c


(vii)

Let Δ=15-2x 11-3x 7-x 11 17 14 10 16 13=015-2x-14+2x 11-3x 7-x 11-28 17 14 10-26 16 13=0 Applying C1C1-2C3 1 11-3x 7-x-17 17 14 -16 16 13 =0 12-3x 4-2x 7-x 0 3 14 0 3 13 =0 Applying C1C1+C2 and C2C2 -C3 12-3x 3×13-3×14 =012-3x-3=012-3x =03x =12 x=4

(viii)
Let =11xp+1p+1p+x3x+1x+2=11xppp3x+1x+2 Applying R2R2-R1=p11x1113x+1x+2 =p11x1112x2 Applying R3R3-R1=p01x0112-xx2 Applying C1C1-C2=p2-x×1x11 Expanding along C1=p2-x1-x=0x=1, 2

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