Matrices

Given matrix:

The matrix is:

Goal: Determine the value of for which the matrix is non-singular.

Explanation:

A matrix is non-singular if its determinant is non-zero. So, we need to find the determinant of the given matrix and solve for when the determinant is non-zero.

Step 1: Compute the determinant of the matrix.

We will use cofactor expansion along the first row to find the determinant:

Determinant = =

Step 2: Compute the 2x2 determinants.

Step 3: Substitute these values into the determinant formula.

Determinant =

Determinant =

Step 4: Set the determinant to non-zero for the matrix to be non-singular.

For the matrix to be non-singular, we need the determinant to be non-zero:

2\lambda - 4 \neq 0

Step 5: Solve for .

Conclusion:

The matrix is non-singular for .

We are given the matrix equation:

and

To find , we can multiply both sides of the equation by on the right. This gives:

Since (the identity matrix), this simplifies to:

Now perform the matrix multiplication:

The result of the multiplication is:

The correct value of is:

We are given the equation and the matrix . We need to find matrix .

To solve for , we subtract matrix from both sides of the equation:

Subtracting corresponding elements of the matrices:

The matrix is .

We are asked to calculate the product . The matrix is a 1x3 matrix and is a 3x2 matrix. The product will be a 1x2 matrix. To calculate the elements of the product, perform the matrix multiplication: For the first element of the resulting matrix: For the second element of the resulting matrix: Thus, the product matrix is

We are given the matrix and the matrix product . We need to find , the determinant of the transpose of matrix .
Step 1: Use the property of determinants for matrix multiplication We know that: and Thus, to find , we first need to calculate and then .
Step 2: Calculate and First, calculate the determinant of matrix : Now, calculate the determinant of matrix :
Step 3: Find From the equation , we have: Thus:
Step 4: Conclusion Since , we have:

We know that the adjugate of a matrix , denoted , is related to the inverse of by the following formula: Since , we can find as follows:

The correct option is (B) :

Given: The adjoint of matrix is .

We know the relationship between a matrix and its adjugate: where is the cofactor matrix of .

First compute the cofactor matrix :

Thus the adjugate matrix is:

Now evaluate the expression:

Therefore, the correct answer is 3.

Given is a integer matrix with and .

Since , is an orthogonal matrix. For orthogonal matrices: Given , we have .

For any orthogonal matrix of odd dimension, is always an eigenvalue. This is because: For odd , this implies: