First slide

Applications of determinant

Question

Let $A = [\begin{array}{ll} a b \\ c d \end{array}]$ , be a 2x2 matrix where $a, b, c, d \in {0, 1}$ . The number of such matrices which have inverse is

Moderate

A

5
B

6
C

7
D

8

Solution

$\det (A) = a d - b c$
Note that det(A) can take value –1, 0 or 1. We have
$\det (A) = 1 \Leftrightarrow a d = 1, b c = 0$
$\Leftrightarrow a = 1, d = 1$ or $(b = 0, c = 0, b = 0; c = 1$
$b = 1, c = 0)$
and $\det (A) = - 1 \Leftrightarrow a d = 0$ or $b c = 1$
This is also possible in 3 cases.
$∴ A^{- 1}$ exists in 6 cases.

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