First slide

Combinations

Question

$Let the matrix A = [\begin{array}{ccc} x & y & - z \\ 1 & 2 & 3 \\ 1 & 1 & 2 \end{array}] where x, y, z \in N . If ∣ a d j (adj (adj (a d j A)) | | = 4^{8} 5^{16} then$ $number of such (x, y, z) are$

Moderate

A

28
B

36
C

45
D

55

Solution

$\begin{array}{l} | adj (a d j (adj (adj A))) | = | A |^{16} = 4^{8} \cdot 5^{16} \\ \Rightarrow | A | = 10 \Rightarrow x + y + z = 10 \\ No. of positive integral solution =^{9} C_{2} = 36 \end{array}$

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