Find the values of the variables for which each statement is true, if possible.

Verified step by step guidance

1. Identify the corresponding elements in each matrix. For two matrices to be equal, their elements in the same positions must be equal. For example, if the matrices are \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) and \( \begin{bmatrix} e & f \\ g & h \end{bmatrix} \), then you set up the equations \( a = e \), \( b = f \), \( c = g \), and \( d = h \).
2. Write down the system of equations by equating each corresponding element from the two matrices. This will give you multiple equations involving the variables.
3. Solve the system of equations step-by-step. You can use substitution or elimination methods to find the values of the variables that satisfy all equations simultaneously.
4. Check for consistency in the system. If the system has no solution, then the matrices cannot be equal for any values of the variables. If there is a solution, those values make the matrices equal.
5. Verify your solution by substituting the found values back into the original matrices to ensure that all corresponding elements are indeed equal.

Key Concepts

Matrix Equality

Two matrices are equal if and only if they have the same dimensions and their corresponding entries are equal. For 2x2 matrices, this means each element in the first matrix must be equal to the corresponding element in the second matrix.

Solving Systems of Equations

When matrix entries contain variables, equating corresponding elements forms a system of equations. Solving this system involves finding values of variables that satisfy all equations simultaneously.

Substitution and Simplification

To solve for variables, substitution or algebraic manipulation is used to simplify equations. This process helps isolate variables and find consistent solutions or determine if no solution exists.