First slide

Applications of determinant

Question

The number of 2 × 2 matrices $A = [\begin{array}{ll} a b \\ c d \end{array}]$ or which ${[\begin{array}{ll} a b \\ c d \end{array}]}^{- 1} = [\begin{array}{cc} 1 / a & 1 / b \\ 1 / c & 1 / d \end{array}], (a, b, c, d \in R)$ is

Moderate

A

0
B

1
C

2
D

infinite

Solution

If $a d - b c \neq 0$ then
$A^{- 1} = \frac{1}{a d - b c} [\begin{array}{cc} d & - b \\ - c & a \end{array}]$
Thus, $\frac{d}{a d - b c} = \frac{1}{a} \Leftrightarrow a d = a d - b c$
$\Leftrightarrow b c = 0 \Leftrightarrow b = 0$ or $c = 0$
Therefore,
${[\begin{array}{ll} a b \\ c d \end{array}]}^{- 1} = [\begin{array}{ll} 1 / a 1 / b \\ 1 / c 1 / d \end{array}]$
is never possible.

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