Assertion: If $A [\begin{array}{ll} 2 3 \\ 1 2 \end{array}] and B [\begin{array}{cc} 2 & - 3 \\ - 1 & 2 \end{array}]$ , then B is the inverse of A
Reason: If A is a square matrix of order m and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse of A.

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Both A and R are true and R is the correct explanation of A.

Both A and R are true and R is not the correct explanation of A.

A is true but R is false.

A is False but R is true.

answer is A.

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Detailed Solution

$Let A = [\begin{array}{ll} 2 3 \\ 1 2 \end{array}] and B = [\begin{array}{cc} 2 & - 3 \\ - 1 & 2 \end{array}] be two matrices$
$\begin{array}{l} Then, AB = [\begin{array}{ll} 2 & 3 \\ 1 & 2 \end{array}] [\begin{array}{cc} 2 & - 3 \\ - 1 & 2 \end{array}] \\ = [\begin{array}{cc} 4 - 3 & - 6 + 6 \\ 2 - 2 & - 3 + 4 \end{array}] = [\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}] = I \end{array}$
$\begin{array}{l} Also, BA = [\begin{array}{cc} 2 & - 3 \\ - 1 & 2 \end{array}] [\begin{array}{ll} 2 & 3 \\ 1 & 2 \end{array}] \\ = [\begin{array}{cc} 4 - 3 & 6 - 6 \\ - 2 + 2 & - 3 + 4 \end{array}] \\ = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = I \end{array}$
Thus, B is the inverse of A.
Both A and R are true and R is the correct explanation of A.