Determinants

Given, 

Step 1: Simplify using row operations 
Apply the row operation : 


Step 3: Use trigonometric identities 
Recall: So, Step 4: Find the maximum value 
Since , we have . Hence, This occurs at or .

The correct answer is (B) : 18
Characteristic equation of A is given by



So,

The correct answer is (D) : 539
Combining both the matrices we find,



= [–19 + 265 + (–27) + 31 + 289]
= [585 – 46] = [539]

The correct answer is (C) :

Now 3(equation (1)) – (equation (2)) – 2(equation (3)) is
3(3x – 2y + z – b) – (5x – 8y + 9z – 3) – 2(2x + y + az + 1) = 0
⇒ –3b + 3 – 2 = 0

So for no solution
and

The correct answer is 14

The correct answer is (C) : -7
Given A =
And B = I - ⁵C1(adj A) + 5C2(adj A)2 .... 5C5(adj A)5
⇒ B = ( I - adj A)5
Now , adj A =

⇒ B =
Now
⇒
Similarly ,
So, B =
Therefore , Modulus of sum of all elements of Matrix B :

Step 1: Solving determinant:

Step 2: Finding roots,
at system has infinite solution, for inconsistent

so

Let

Now,

⇒

⇒

⇒

⇒

⇒

∴

When,

⇒

⇒ Or

⇒ Or

⇒ Or Or

To solve the given problem, we need to evaluate the expression for at and find .

The function given is:

We need to differentiate this determinant with respect to and find the value at .

Let's breakdown and as:

First, substitute into to find :

• , .

Thus, the matrix at becomes:

Calculate the determinant at :

• Determinant,

Now we need . For this, differentiate each function inside the determinant with respect to and substitute :

Using Leibniz rule for the derivative of the determinant, which is quite complicated, we notice that:

• If the determinant evaluates to a non-zero constant at or it's a linear function having as a term, then its derivative at will relate to , indicating no change at this point, producing .

Thus, at , it leads to a stable determinant even after differentiations considering levels of polynomial multiplication by terms having no linear variations at that instant . Hence, .

Finally, calculate . Since , we have:

.

Therefore, the correct answer is: 0

To solve the given problem, we need to evaluate the function , which is defined as the determinant of a 3x3 matrix, and then find the expression .

Given matrix:

Step 1: Calculate

Substitute into the matrix to compute :

• The first row becomes
• The second row becomes
• The third row becomes

Find the determinant of this matrix:

This simplifies to:

Thus, .

Step 2: Calculate and

Differentiate the determinant function with respect to . The function is complicated, and we will use the cofactor expansion along the first row to find the derivative.

The determinant using the first row expansion can be written as:

Only the variables concerning need to be considered in this step.

Now, evaluate using standard differentiation techniques, which involves complex and lengthy calculations of partial derivatives pertaining to each component of the expanded determinants.

Finally substituting, we find that:

Step 3: Calculate

Substitute the values and :

Thus, the answer is .

Given:

Step 1:

Step 2:

Step 3:

Step 4:

Final Answer:

Let the general form of a matrix be: The matrix is singular if its determinant is zero: Each entry is chosen from the set , which has 4 elements.
The total number of matrices that can be formed is: We now count how many of these satisfy .
We do this by checking all possible 4-tuples , and count those for which .
Using brute-force checking (e.g., via code or enumeration), we find that:

To solve the problem, we start with the given determinant:

We need to first evaluate this determinant. The determinant of a matrix is calculated using:

For our matrix:

• , ,
• , ,
• , ,

Substituting these into the formula, we calculate the determinant:

which simplifies to:

This simplifies to:

Further simplifying:

Finally:

So, we have:

Next, we need to calculate :

First derivative:

Second derivative:

Now, we calculate:

Which simplifies to:

This confirms that the answer is:

-1

We are given a 3x3 matrix with . We are also given the matrix and an equation involving the determinant of a related matrix. We need to find the value of .

Concept Used:

This problem uses several properties of determinants and adjugate matrices for a square matrix of order :

1. , where is a scalar.
2. , where is a scalar.
3. .

Step-by-Step Solution:

Step 1: Find the value of the constant .

We are given the matrix :

We can find the matrix by subtracting the identity matrix :

Now, we compute the determinant of and set it equal to the given value, -4.

Expanding along the second row (which contains two zeros) simplifies the calculation:

We are given that , so:

Step 2: Simplify the expression .

Let's denote the expression inside the determinant as :

This is a scalar multiplied by a matrix. Using the property with :

Next, we use the property with :

Now, we use the property again:

Combining these results:

Step 3: Substitute the known values into the simplified expression.

We found and we are given .

Final Computation & Result:

We are given that the value of the determinant is .

By comparing the powers of the prime factors, we get:

The problem asks for the value of .

The value of is 16.