First slide

Combinations

Question

Let $S = \{[\begin{array}{cc} 18 & a \\ b & 25 \end{array}] : a, b \in N\}$ The number of singular matrices in $s$ is

Moderate

A

18
B

25
C

450
D

infinite

Solution

det $[\begin{array}{cc} 18 & a \\ b & 25 \end{array}] = 2 \times 3^{2} \times 5^{2} - a b = 0$
$\begin{array}{lrl} \Leftrightarrow a b = 2 \times 3^{2} \times 5^{2} \\ \Leftrightarrow a ∣ 2 \times 3^{2} \times 5^{2} \end{array}$
That is $a$ is a divisor of $2 \times 3^{2} \times 5^{2}$
Therefore, $a$ is of the form $2^{a} 3^{b} 5^{2}$ where
$α \in {0, 1}, β, γ \in {0, 1, 2}$ .
Thus number of elements in $S is 2 \times 3 \times 3 = 18 .$

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