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Standard XII

Mathematics

Singular and Non Singualar Matrices

Let A=[ a b...

Question

Let $A = [\begin{matrix} a & b c & d \end{matrix}]$ be a $2 \times 2$ matrix, where a, b, c and d take the values $0$ or $1$ only. The number of such matrices which have inverses is?

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Solution

The correct option is D $8$
$A^{- 1} = \frac{a d j A}{| A |}$
so $| A | \neq 0 a d - b c \neq 0$
for $a d - b c \neq 0$
either $a d = 1, b c = 0$ or $a d = 0, b c = 1$
$a d = 1, b c = 0$ can be chosen in 4 ways
similarly $a d = 0, b c = 1$ can be chosen in 4 ways
so number of matrices formed $= 4 + 4 = 8$

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Let A=[abcd] be a 2×2 matrix, where a, b, c and d take the values 0 or 1 only. The number of such matrices which have inverses is?

The correct option is D 8A−1=adjA|A|so |A|≠0ad−bc≠0for ad−bc≠0 either ad=1,bc=0 or ad=0,bc=1ad=1,bc=0 can be chosen in 4 ways similarly ad=0,bc=1 can be chosen in 4 waysso number of matrices formed=4+4=8

Let $A = [\begin{matrix} a & b c & d \end{matrix}]$ be a $2 \times 2$ matrix, where a, b, c and d take the values $0$ or $1$ only. The number of such matrices which have inverses is?