If \(\begin{bmatrix} x+y+z+w\\ x+y\\ x+z\\ x+w\end{bmatrix} =\begin{bmatrix}10\\ 6\\ 5\\ 3\end{bmatrix}\), find the value of \(x^2+y^2-(w+z)^2\).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4
By equality of matrices, corresponding entries are equal:
\(x + y + z + w = 10 \quad (1)\)
\(x + y = 6 \quad (2)\)
\(x + z = 5 \quad (3)\)
\(x + w = 3 \quad (4)\)
Adding (2), (3) and (4):
\(3x + y + z + w = 14\)
Subtracting (1):
\(2x = 4,\quad x = 2\)
\(y = 4,\quad z = 3,\quad w = 1\)
Then
\(x^2 + y^2 - (w + z)^2 = 4 + 16 - (1 + 3)^2\)
\(= 20 - 16 = 4\)