Matrices

For orthogonal matrices: M⁻¹(θ)=M(-θ) Also: (AB)⁻¹=B⁻¹A⁻¹ Hence: [M(α)M(β)]⁻¹=M(-β)M(-α)

There is no solution when the matrix is inconsistent.
The solution of a system of equations is not defined when the system is inconsistent or the equations are dependent. Let's discuss each case in the context of matrices.
Inconsistent system: An inconsistent system of equations has no solution. In matrix form, this occurs when the system of equations can be represented as an augmented matrix [A|B], where A is the coefficient matrix and B is the constant matrix. If the row-reduced echelon form (RREF) of [A|B] contains a row of the form [0 0 ... 0 | b], where b is a nonzero constant, then the system is inconsistent and has no solution.
Dependent equations: Dependent equations occur when one equation can be obtained by adding or multiplying other equations in the system. In matrix form, this means that the rows of the augmented matrix [A|B] are linearly dependent. To determine linear dependence, we can row-reduce the augmented matrix to its RREF. If the RREF contains a row of the form [0 0 ... 0 | 0], where the corresponding row in the coefficient matrix A is not all zeros, then the system has dependent equations. In this case, there are infinitely many solutions, as the equations are not providing independent information.
To summarize, the solution of a system of equations is not defined when:
The system is inconsistent, and the augmented matrix [A|B] in RREF form contains a row of the form [0 0 ... 0 | b], where b is nonzero. The equations are dependent, and the augmented matrix [A|B] in RREF form contains a row of the form [0 0 ... 0 | 0], where the corresponding row in the coefficient matrix A is not all zeros.

Step 1: {Expand both determinants}
Step 2: {Solve for }
Step 3: {Compute }
Since , Step 4: {Conclusion}

Step 1: {Use the determinant property of adjugate matrices}
For a square matrix of order , the determinant of its adjugate is given by: Since is of order 3, we get: Step 2: {Simplify further}
Step 3: {Compute }
Step 4: {Conclusion}
Thus, the correct answer is .

Step 1: {Show that is an orthogonal matrix}
Since , the matrix is orthogonal.
Step 2: {Simplify }
Step 3: {Compute }
Step 4: {Conclusion}

Step 1: Determinant of matrix
For a 2x2 matrix , the determinant is . For: Since is diagonal, the determinant is the product of diagonal elements: .

Step 2: Determinant of
Using the property of determinants, for any square matrix : So: Calculate:
Step 3: Alternative approach
Compute : Determinant of : Both methods agree.

Step 4: Check options
Option (1) 216 matches our result.

The determinant of a matrix is given by: For , Now, the determinant of is: The correct answer is:

The determinant of a 2x2 matrix is given by: For the matrix , we have: Thus, the determinant of the matrix is: