Matrices And Determinants

The determinant is:

Step 1: Factor out common terms.

Factor out from each row:

Step 2: Simplify the remaining determinant.

The simplified determinant is:

Thus:

Step 3: Match with the given expression.

The given determinant is:

Rewrite:

The full expression becomes:

Step 4: Equate powers of x, y, z.

Compare with

This implies k = -1

This implies:

Conclusion: The value of k is:

.

The given matrix equation is:

Let Substituting A into the equation, we compute step-by-step.

Step 1: Simplify the right product

Multiply the matrices:

Step 2: Multiply by

Now multiply:

Perform the matrix multiplication:

Simplify each term: - First row, first column:

- First row, second column:

- Second row, first column:

- Second row, second column:

Thus, the resulting matrix is:

Step 3: Equate to identity matrix.

We equate:

From this, solve the system of equations:

Solve this system, and you find:

Conclusion: The matrix A is: