Determinants

Given determinant:

To find: Find the value of d.

Step 1: Compute the determinant of the 3x3 matrix.

The determinant of the given matrix can be computed using cofactor expansion along the first row:

Step 2: Compute each 2x2 determinant.

• For the first 2x2 matrix:
• For the second 2x2 matrix:
• For the third 2x2 matrix:

Step 3: Substitute these values back into the determinant expression.

Step 4: Expand and simplify.

First, expand each term:

-2(2x - 4) = -4x + 8

1(x + 5) = x + 5

Now combine all terms:

Simplify the expression:

Step 5: Identify the value of d.

The given determinant is equal to , where , , , and .

Answer: The value of d is 16.

We are given the determinant of the matrix:

To solve for , we will expand the determinant along the first row:

Now, calculate the 2x2 determinants:

Now substitute these values back into the determinant expansion:

Set this equal to 0 to solve for :

Factor the quadratic equation:

Thus, the solutions are:

or

The values of are 5 and -3.

We are asked to find the value of the determinant:

This is a 3x3 matrix where the elements are powers of 4, 3, and 2. We can calculate the determinant by expanding along the first row:

Now, calculate the 2x2 determinants:

Substitute these values back into the determinant expansion:

The value of the determinant is .

We are given the matrix :

and we need to find , where:

Step 1: Understand the relationship between and

We know the following property of determinants: And for a 3x3 matrix, the determinant of the adjugate matrix is related to by the formula: where is the order of the matrix. For a 3x3 matrix, , so: Thus:

Step 2: Calculate the determinant of

To find , we use the formula for the determinant of a 3x3 matrix: We calculate each 2x2 determinant: Now substitute these values back into the determinant formula:

Step 3: Calculate

Now that we know , we can calculate :

The correct option is (D) :

We are tasked with evaluating the determinant of the following matrix: We will use the property of determinants: the determinant of a matrix with two proportional rows or columns is zero. To check if the rows or columns are proportional, consider the following:
The first column has the values .
The second column has .
The third column has . Notice that the second and third columns have a form of similarity due to the presence of trigonometric identities.
In particular, the expressions and have a linear relation due to the angle addition formula for cosine.
This similarity suggests that the rows are linearly dependent, leading to a determinant of zero.

The correct option is (E) :

We need to calculate the determinant of the matrix: Using cofactor expansion along the first row: We calculate the determinant: Thus, the determinant becomes: We set the determinant equal to 0: Solving this gives:

The correct option is (E) :

Given matrix and the condition .

First, compute and :

Compute :

Compute :

Set :

This gives the system of equations: 1. 2. 3.

From equation (2): Substitute into equation (3): Then

Now check the options: (A) → False (B) → False (C) → False (D) is not symmetric → False (E) → True

The correct statement is (E).

We need to compute the determinant of:

Using the formula for a 3×3 determinant: where the matrix is:

Calculating:

Therefore, the determinant is 52.