Problem 40 - Algebra 2

Question

Solve each matrix equation. \(\left[\begin{array}{ll}{x} & {y}\end{array}\right]=\left[\begin{array}{ll}{y} & {4}\end{array}\right]\)

Step-by-Step Solution

Answer

x = 4, y = 4

Step 1: Set Up the Equation

Given the matrix equation \(\left[\begin{array}{ll} x & y \end{array}\right] = \left[\begin{array}{ll} y & 4 \end{array}\right]\), identify the corresponding elements that need to be equal. This gives us two equations: \(x = y\) and \(y = 4\).

Step 2: Solve for y

From the second equation \(y = 4\), we find that \(y\) is equal to 4.

Step 3: Substitute Back to Find x

Substitute \(y = 4\) into the first equation \(x = y\). Since \(y = 4\), we substitute to find \(x = 4\).

Step 4: Verify the Solution

Check that the solution \(x = 4\) and \(y = 4\) satisfies both original equations of the matrix. Substitute these values back: \(\left[\begin{array}{ll} 4 & 4 \end{array}\right] = \left[\begin{array}{ll} 4 & 4 \end{array}\right]\), which confirms both sides are equal.

Key Concepts

Solving Matrices

Solving matrix equations is like solving puzzles; each piece must fit perfectly where it belongs.
In this exercise, we are given a matrix equation where two matrices are set equal to each other: \[ \begin{array}{ll} [x & y] = [y & 4] \end{array} \] The goal here is to identify what values of \(x\) and \(y\) will make the two sides identical.
Unlike regular algebraic equations, each part of one matrix must correspond directly to the same part in the other matrix. So, solving matrices involves breaking down the larger equation into several smaller ones, focusing on individual elements.
It's almost like breaking a big question into manageable bits, making it easier to solve. The core of solving this matrix involves finding values that satisfy all these smaller equations simultaneously.

Element-wise Equality

A matrix equation relies on element-wise equality, meaning each element in one matrix must equal the corresponding element in the other.
In our given equation, for instance, the first element \(x\) in the first matrix must equal the first element \(y\) in the second matrix. Similarly, the second element \(y\) in the first matrix must be equal to the second element \(4\) in the second matrix.
Forming these small equations, which in our case are \(x = y\) and \(y = 4\), is crucial. It allows us to methodically solve each equation to find values for unknowns like \(x\) and \(y\). By satisfying this element-wise relationship, we ensure the entire matrix equation holds true.

Matrix Verification

Once you find what seems like the right solution, the final step in solving matrices is verification.
Verification means double-checking your work, ensuring each solution correctly satisfies the original equations.
Using our values \(x = 4\) and \(y = 4\), we verify them by substituting back into the original matrix equation: \[ \begin{array}{ll} [4 & 4] = [4 & 4] \end{array} \] The left side equals the right side, confirming that our solutions are accurate! Verification is a habit that ensures accuracy.
It prevents errors from creeping in, making it a vital part of solving any mathematical problem, especially matrix problems.