Kerala KEAM 5 June 2024 Solved Paper

We are given that: $$XA = B.$$To solve for $X$, we multiply both sides by the inverse of $A$ on the right: $$X = B A^{-1}.$$Step 1: Find the inverse of $A$. The formula for the inverse of a 2x2 matrix $A = \begin{pmatrix} a & b < br>c & d \end{pmatrix}$ is: $$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b < br>-c & a \end{pmatrix}.$$For $A = \begin{pmatrix} 0 & 1-1 & 2 \end{pmatrix}$, we have: $$Determinant of A = (0)(2) - (1)(-1) = 1.$$So, the inverse is: $$A^{-1} = \begin{pmatrix} 2 & -1 < br>1 & 0 \end{pmatrix}.$$Step 2: Now, calculate $X = B A^{-1}$: $$B = \begin{pmatrix} 1 & 1 < br>-1 & -1 \end{pmatrix}, \quad A^{-1} = \begin{pmatrix} 2 & -1 < br>1 & 0 \end{pmatrix}.$$Multiply $B$ and $A^{-1}$: $$X = \begin{pmatrix} 1 & 1 < br>-1 & -1 \end{pmatrix} \begin{pmatrix} 2 & -1 < br>1 & 0 \end{pmatrix} = \begin{pmatrix} (1)(2) + (1)(1) & (1)(-1) + (1)(0) < br>(-1)(2) + (-1)(1) & (-1)(-1) + (-1)(0) \end{pmatrix}.$$ $$X = \begin{pmatrix} 3 & -1 < br>-3 & 1 \end{pmatrix}.$$Thus, the value of $X$ is $\begin{pmatrix} 3 & -1 < br>-3 & 1 \end{pmatrix}$.Therefore, the correct answer is option (C).Quick Tip: When solving matrix equations of the form $XA = B$, you can isolate $X$ by multiplying both sides by $A^{-1}$ from the right. Be sure to compute the inverse correctly using the determinant and the adjoint matrix.