The teacher hasn't uploaded a solution for this question yet.
We are given a matrix $A$ and its inverse $A^{-1}$. We need to find the values of $a, x, b,$ and $y$ and then compute $(a+x)-(b+y)$. We know that $A \cdot A^{-1} = I$, where $I$ is the identity matrix.
We multiply the given matrix $A$ and $A^{-1}$: $$A \cdot A^{-1} = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
Let's calculate the elements of the resulting matrix by performing the matrix multiplication: \r\nElement (1,1): $(-1)(1) + a(-8) + 2(b) = 1 \implies -1 - 8a + 2b = 1 \implies -8a + 2b = 2 \implies -4a + b = 1$ (1)\r\nElement (1,2): $(-1)(-1) + a(7) + 2(y) = 0 \implies 1 + 7a + 2y = 0 \implies 7a + 2y = -1$ (2)\r\nElement (1,3): $(-1)(1) + a(-5) + 2(3) = 0 \implies -1 - 5a + 6 = 0 \implies -5a = -5 \implies a = 1$\r\nElement (2,1): $(1)(1) + 2(-8) + x(b) = 0 \implies 1 - 16 + bx = 0 \implies bx = 15$ (3)\r\nElement (2,2): $(1)(-1) + 2(7) + x(y) = 1 \implies -1 + 14 + xy = 1 \implies xy = -12$ (4)\r\nElement (2,3): $(1)(1) + 2(-5) + x(3) = 0 \implies 1 - 10 + 3x = 0 \implies 3x = 9 \implies x = 3$\r\nElement (3,1): $(3)(1) + 1(-8) + 1(b) = 0 \implies 3 - 8 + b = 0 \implies b = 5$\r\nElement (3,2): $(3)(-1) + 1(7) + 1(y) = 0 \implies -3 + 7 + y = 0 \implies y = -4$\r\nElement (3,3): $(3)(1) + 1(-5) + 1(3) = 1 \implies 3 - 5 + 3 = 1 \implies 1 = 1$ (This is consistent)\r\n
From Element (1,3), we found $a = 1$. \r\nFrom Element (2,3), we found $x = 3$. \r\nFrom Element (3,1), we found $b = 5$. \r\nFrom Element (3,2), we found $y = -4$. \r\nLet's verify these values with the other equations:\r\nFrom (1): $-4a + b = 1 \implies -4(1) + 5 = 1 \implies 1 = 1$ (Correct)\r\nFrom (2): $7a + 2y = -1 \implies 7(1) + 2(-4) = -1 \implies 7 - 8 = -1 \implies -1 = -1$ (Correct)\r\nFrom (3): $bx = 15 \implies 5(3) = 15 \implies 15 = 15$ (Correct)\r\nFrom (4): $xy = -12 \implies 3(-4) = -12 \implies -12 = -12$ (Correct)
Now we can calculate $(a+x)-(b+y)$: \r\n$(a+x)-(b+y) = (1+3) - (5+(-4)) = 4 - (5-4) = 4 - 1 = 3$
Final Answer: 3
AI generated content. Review strictly for academic accuracy.