Question

Let \(A = [\begin{array}{cc}3 & -4 \\ 1 & -1\end{array}]\) and \(B = [\begin{array}{cc}6 & -13 \\ 5 & -10\end{array}]\) be two matrices. If the variables \(x\) and \(y\) satisfy the matrix equation \(((A^{-1})^2+B)[\begin{array}{c}x \\ y\end{array}] = [\begin{array}{c}0 \\ 0\end{array}]\), then the ordered pair \((x,y) =\)

• A. \((3,5)\)
• B. \((10,7)\)
• C. \((4,6)\)
• D. \((5,3)\)

Hint:

Compute the inverse of A, square it, add B and solve the resulting homogeneous system.

Answer 1

The Correct Option is D

Step 1: Alternative check:
Compute $M\begin{pmatrix} x \\ y \end{pmatrix}$ for each option by using $M$ found from $A^{-1}$.

Step 2: Test:
$M = \begin{pmatrix} 3 & -5 \\ 3 & -5 \end{pmatrix}$ gives $3x - 5y$.
For $(5,3)$ this is $15 - 15 = 0$.
For the others it is $-16$, $-5$, $-18$, so only $(5, 3)$ works.

Final Answer:
The ordered pair is $(5, 3)$, option (D). \[ \boxed{(5,3)} \]