Matrices And Determinants

We are given that and where is the transpose of matrix , and is the identity matrix. First, find the transpose of : Now, add and : We are told that , so: This implies that: Solving for : Thus, , which gives:
Conclusion: The value of is .

We are given the system of equations:

Step 1: Solve for and in terms of .
From Equation 1: From Equation 2:

Step 2: Substitute and into Equation 3.
Substitute and into Equation 3: Simplify the equation:

Step 3: Find and .
Now that we know , substitute this value into the expressions for and : Conclusion: The values of , , and are:

The adjugate (adjoint) of a matrix , denoted by , is the transpose of the cofactor matrix of . A well-known result in matrix theory is that: To prove this, we first calculate , the determinant of . The determinant of a 3x3 matrix is given by: Using cofactor expansion along the first row: Now, calculate the 2x2 determinants: Substitute these into the determinant formula: Thus, . Since the determinant is 1, we have:
Conclusion: We have proved that

We need to prove that , and also find .
Step 1: Find the Determinant of The determinant of matrix is given by: Expanding along the first row: Now, compute the 2x2 determinants: Substitute these into the determinant formula: Thus, .
Step 2: Verify the Formula The adjugate (adjoint) of , denoted by , is the transpose of the cofactor matrix of . Since , we have: Thus, , proving the required result.
Step 3: Find the Inverse of The inverse of is given by: Since , we have: So, we need to find the adjugate matrix , which is the transpose of the cofactor matrix of . The cofactor matrix is calculated by finding the cofactor of each element in , and then taking the transpose. The cofactor matrix is: The adjugate matrix is the transpose of : Thus, the inverse of is:
Conclusion: We have proved that and the inverse of is

We can solve this system of linear equations using the matrix method.
Step 1: Write the system as a matrix equation We write the system of equations as: Let , , and . The matrix equation is .
Step 2: Find We already computed in part (a), and it is:
Step 3: Solve for Now, solve for by multiplying both sides of the equation by : Substitute the values: Performing the matrix multiplication: Thus, the solution to the system of equations is:
Conclusion: The solution to the system of equations is:

Step 1: Begin by expanding the determinant along the first row: Step 2: Now calculate the 2×2 minors: Step 3: Now substitute these back into the determinant formula: Step 4: Simplify the above expression, carefully combining like terms: After simplifying, we arrive at: Thus, we have proved the identity.

Step 1: Expand the determinant: Step 2: Calculate the 2x2 determinants: Step 3: Substitute the results back: Step 4: Now, set the determinant equal to zero and simplify. After solving the equation, we obtain the result .