- Perform component-wise matrix addition: \(\begin{bmatrix} x^2 - 4 \ 2y - 4 \end{bmatrix} = \begin{bmatrix} 32 \ y + 6 \end{bmatrix}\).
- Form a system of equations: \(x^2 - 4 = 32\) and \(2y - 4 = y + 6\).
- Solve \(x^2 = 36\) to get \(x = 6\) or \(x = -6\).
- Solve \(y - 4 = 6\) to get \(y = 10\).
- The solution sets are \(x = \boxed{6, -6}\) and \(y = \boxed{10}\).
Explanation
- Perform Matrix Addition and Set Up Equations
To solve the matrix equation \(\begin{bmatrix} x^2 \ y \end{bmatrix} + \begin{bmatrix} -4 \ y-4 \end{bmatrix} = \begin{bmatrix} 32 \ y+6 \end{bmatrix}\), we begin by performing matrix addition on the left-hand side. The addition of two matrices is performed by adding their corresponding elements. Applying this rule, we sum the vectors as follows:
\[\begin{bmatrix} x^2 + (-4) \ y + (y-4) \end{bmatrix} = \begin{bmatrix} x^2 - 4 \ 2y - 4 \end{bmatrix}\]
Now, the equation is:
\[\begin{bmatrix} x^2 - 4 \ 2y - 4 \end{bmatrix} = \begin{bmatrix} 32 \ y+6 \end{bmatrix}\]
- Solve the System of Equations
Since two matrices are equal if and only if their corresponding components are equal, we can form a system of two equations:
- For the first component: \(x^2 - 4 = 32\)
- For the second component: \(2y - 4 = y + 6\)
We solve each equation individually. Starting with the first one, adding 4 to both sides gives:
\[x^2 = 36\]
Taking the square root of both sides, we find two possible values for \(x\):
\[x = 6 \quad \text{or} \quad x = -6\]
Next, we solve for \(y\) in the second equation:
\[2y - 4 = y + 6\]
Subtracting \(y\) from both sides gives \(y - 4 = 6\), and adding 4 to both sides results in:
\[y = 10\]
- Conclusion and Final Results
We have found that for the matrix equation to hold, \(x\) must be either \(6\) or \(-6\), and \(y\) must be \(10\). Thus, the possible solutions are \((x, y) = (6, 10)\) and \((x, y) = (-6, 10)\). The final values are \(x = \pm 6\) and \(y = 10\).
Examples
In engineering and physics, matrices are often used to represent transformations or physical states that evolve over time. For example, if a system's state is described by a vector \(\mathbf{v} = [x^2, y]^T\), and it undergoes an adjustment by \(\mathbf{a} = [-4, y-4]^T\), the new state \(\mathbf{v}' = [32, y+6]^T\) allows you to work backwards to solve for the original parameters. Mastering this helps in tasks like debugging sensor readings or adjusting control variables in automated machinery.