First slide

Introduction to Determinants

Question

Let $A = (\begin{array}{ll} a b \\ c d \end{array})$ , where $a, b, c, d \in R$ Then

Moderate

A

$\det (A) \leq \sqrt{a^{2} + b^{2}} \sqrt{c^{2} + d^{2}}$
B

$\det (A) \leq (a + b) (c + d)$
C

$\det (A) \leq a c + b d$
D

$\det (A) \leq (| a | - | b |) (| c | - | d |)$

Solution

$\begin{array}{l} \det (A) = a d - b c \leq | a d - b c | \\ \leq | a ∥ d | + | b ∥ c | \end{array}$
Now,
$\begin{array}{l} (a^{2} + b^{2}) (c^{2} + d^{2}) - (| a ∥ d | + | b ∥ c |)^{2} \\ = a^{2} c^{2} + b^{2} c^{2} + a^{2} d^{2} + b^{2} d^{2} \\ - (a^{2} d^{2} + b^{2} c^{2} + 2 | a ∥ d ∥ b ∥ c |) \\ = (| a c | - | b d |)^{2} \geq 0 \end{array}$
$\Rightarrow | a ∥ d | + | b | | c | \leq \sqrt{a^{2} + b^{2}} \sqrt{c^{2} + d^{2}}$

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