Question

If \( A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} \) and \( A^{-1} = \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} \), find the value of \( (a + x) - (b + y) \).

Hint:

To solve problems involving matrix inverses, use the property \(A \cdot A^{-1} = I\), and analyze the resulting equations row by row. Simplify systematically to find unknown elements.

Answer 1

Step 1: Use the property of matrix inverses.

The product of a matrix \( A \) and its inverse \( A^{-1} \) is the identity matrix:

\[ A \cdot A^{-1} = I_3, \] where \( I_3 \) is the \( 3 \times 3 \) identity matrix.

Step 2: Multiply \( A \) and \( A^{-1} \).

Compute the product \( A \cdot A^{-1} \):

\[ \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}. \]

Step 3: Analyze each element of the product.

From the first row of the product:

\[ [-1(1) + a(-8) + 2(b)] = 1, \quad [-1(-1) + a(7) + 2(y)] = 0, \quad [-1(1) + a(-5) + 2(3)] = 0. \]

Simplify each equation:

1. \(-1 - 8a + 2b = 1 \) implies \(-8a + 2b = 2 \) implies \(4a - b = -1. \)
2. \(1 + 7a + 2y = 0\) implies \(7a + 2y = -1. \)
3. \(-1 - 5a + 6 = 0 \) implies \(-5a = -5 \) implies a = 1.

From the third row of the product:

\[ [3(1) + 1(-8) + 1(b)] = 0, \quad [3(-1) + 1(7) + 1(y)] = 0, \quad [3(1) + 1(-5) + 1(3)] = 1. \]

Simplify each equation:

1. \(3 - 8 + b = 0 \) implies b = 5.
2. \(-3 + 7 + y = 0 \) implies y = -4.

Step 4: Compute \((a + x) - (b + y)\).

Substitute the values \( a = 1 \), \( x = 3 \), \( b = 5 \), and \( y = -4 \):

\[ (a + x) - (b + y) = (1 + 3) - (5 + (-4)). \]

Simplify:

\[ (a + x) - (b + y) = 4 - (5 - 4) = 4 - 1 = 3. \]

Final Answer:

\[ (a + x) - (b + y) = 3. \]