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Algebra> determinants...

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Let the three digit numbers A28,3B9,62C where A, B, C are integers between 0 and 9 , be divisible by a fixed integer k. Show that the determinant

A 3 6
8 9 C
2 B 2
is divisible by k

Gurmandeep Singh Sootdhar , 16 Years ago

Grade 11

1 Answers

SHAIK AASIF AHAMED

Hello student,
Please find the answer to your question below
AsA28,3B9,62C where A, B, C are integers between 0 and 9 , be divisible by a fixed integer k then there exist m1,m2,m3 such that
100A+20+8=km1
300+10B+9=km2
600+20+c=km3
Applying R2=R2+100R1+10R2 we get
$\Delta =\begin{bmatrix} A & 3 &6 \\ km1&km2 &km3 \\ 2& B & 2 \end{bmatrix}=kM$
Where $M=\begin{bmatrix} A & 3 &6 \\ m1&m2 &m3 \\ 2& B & 2 \end{bmatrix}$ is an integer
Since A,m1,m2,m3,B are integers
Thus $\Delta$ is divisible by k

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