Matrix

We are given the matrix as:
To find , we can use the matrix inverse formula or perform elementary row operations. However, in this case, based on the structure of the matrix, we observe that the solution involves manipulating and its powers. Using the properties of matrix operations, the inverse of the matrix is given by the expression . Thus, .

We are given the matrix as:
To determine the rank of the matrix, we can perform row operations to reduce the matrix to its row echelon form. Step 1: Subtract 2 times the first row from the second row: Step 2: Subtract the first row from the third row: Step 3: Multiply the second row by and the third row by : Step 4: Subtract the third row from the first row: Now, the matrix is in row echelon form, and we can see that there are two non-zero rows. Hence, the rank of the matrix is 2. Thus, the rank of matrix is 2.

Step 1: Understand the decomposition
Where:
is symmetric
is skew-symmetric
Using the standard identity for decomposing any square matrix into symmetric and skew-symmetric parts:
Step 2: Observe what's asked
We are asked to find . But since is symmetric and is skew-symmetric, we know: Hence, the correct answer is:

We are given the matrix: To find the determinant of this matrix, we use cofactor expansion along the first row: After performing the calculations for each of the 2x2 determinants and simplifying the terms, we find that: Thus, the value of the determinant is .

Step 1: Solving for matrix . Given that , we solve for : Step 2: Use the condition . We compute and set it equal to the identity matrix, which gives us the relationships between and . Step 3: Solving for and . From the equations, we find that and . Step 4: Compute .

We are given that is a matrix and , and we are to determine if . - From the reason (R), we know that for any square matrix of order , the determinant of its adjugate matrix is given by: - For being a matrix ( ), we apply the formula: So, the assertion (A) that is false. Thus, (A) is false but (R) is true.

Given: Step 1: Formula for Determinant
For a 2x2 matrix , the determinant is: Step 2: Substitute the Values
Identify , , , . Compute: Thus, the determinant is: