Given that,Determine the 2 × 2 matrices X and Y satisfyingi. AX = B;ii. Y A = B. (6)

To determine the $2 \times 2$ matrices $X$ and $Y$ that satisfy the equations $AX = B$ and $YA = B$, we will proceed through the following steps:

1. Break Down the Problem

1. Identify matrices $A$, $B$, and the unknown matrices $X$ and $Y$.
2. We need to solve two separate matrix equations for $X$ and $Y$.

2. Relevant Concepts

1. The equation $AX = B$ can be solved for $X$ by multiplying both sides by the inverse of $A$ (if $A$ is invertible): $X = A^{-1}B$

2. The equation $YA = B$ can similarly be solved for $Y$: $Y = BA^{-1}$

3. Analysis and Detail

• For both equations to be solved, we must ensure that the matrix $A$ is invertible. This requires the determinant of $A$, $\det(A)$, to be non-zero.

Calculating $X$ and $Y$:

1. Compute the inverse of matrix $A$, $A^{-1}$.
2. Substitute $A^{-1}$ into the equations to find $X$ and $Y$.

4. Verify and Summarize

• After calculating $X$ and $Y$, verify both solutions by substituting them back into the original equations to ensure they hold true.

Final Answer

The final expressions for matrices $X$ and $Y$ are given by: $X = A^{-1}B$ $Y = BA^{-1}$

To provide specific numerical solutions, the actual matrices $A$ and $B$ would need to be provided.